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Theorem el2xptp0 8044

Description: A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)

**Assertion**
Ref	Expression
el2xptp0	⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍)) ↔ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩))

**Proof of Theorem el2xptp0**
Step	Hyp	Ref	Expression
1		xp1st 8029	. . . . . 6 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (1^st ‘𝐴) ∈ (𝑈 × 𝑉))
2	1	ad2antrl 726	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → (1^st ‘𝐴) ∈ (𝑈 × 𝑉))
3		3simpa 1145	. . . . . . 7 ⊢ (((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍) → ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌))
4	3	adantl 480	. . . . . 6 ⊢ ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍)) → ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌))
5	4	adantl 480	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌))
6		eqopi 8033	. . . . 5 ⊢ (((1^st ‘𝐴) ∈ (𝑈 × 𝑉) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌)) → (1^st ‘𝐴) = ⟨𝑋, 𝑌⟩)
7	2, 5, 6	syl2anc 582	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → (1^st ‘𝐴) = ⟨𝑋, 𝑌⟩)
8		simprr3 1220	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → (2^nd ‘𝐴) = 𝑍)
9	7, 8	jca 510	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍))
10		df-ot 4639	. . . . . 6 ⊢ ⟨𝑋, 𝑌, 𝑍⟩ = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩
11	10	eqeq2i 2740	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ 𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩)
12		eqop 8039	. . . . 5 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
13	11, 12	bitrid 282	. . . 4 ⊢ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
14	13	ad2antrl 726	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1^st ‘𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2^nd ‘𝐴) = 𝑍)))
15	9, 14	mpbird 256	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))) → 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩)
16		opelxpi 5717	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
17	16	3adant3 1129	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
18		simp3 1135	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ 𝑊)
19	17, 18	opelxpd 5719	. . . . . 6 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
20	10, 19	eqeltrid 2832	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
21	20	adantr 479	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
22		eleq1 2816	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
23	22	adantl 480	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
24	21, 23	mpbird 256	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → 𝐴 ∈ ((𝑈 × 𝑉) × 𝑊))
25		2fveq3 6905	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1^st ‘(1^st ‘𝐴)) = (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
26		ot1stg 8011	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (1^st ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑋)
27	25, 26	sylan9eqr 2789	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (1^st ‘(1^st ‘𝐴)) = 𝑋)
28		2fveq3 6905	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘(1^st ‘𝐴)) = (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)))
29		ot2ndg 8012	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘(1^st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑌)
30	28, 29	sylan9eqr 2789	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘(1^st ‘𝐴)) = 𝑌)
31		fveq2 6900	. . . . 5 ⊢ (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2^nd ‘𝐴) = (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩))
32		ot3rdg 8013	. . . . . 6 ⊢ (𝑍 ∈ 𝑊 → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
33	32	3ad2ant3 1132	. . . . 5 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (2^nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
34	31, 33	sylan9eqr 2789	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2^nd ‘𝐴) = 𝑍)
35	27, 30, 34	3jca 1125	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍))
36	24, 35	jca 510	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍)))
37	15, 36	impbida 799	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1^st ‘(1^st ‘𝐴)) = 𝑋 ∧ (2^nd ‘(1^st ‘𝐴)) = 𝑌 ∧ (2^nd ‘𝐴) = 𝑍)) ↔ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⟨cop 4636 ⟨cotp 4638 × cxp 5678 ‘cfv 6551 1^st c1st 7995 2^nd c2nd 7996

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-ot 4639 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-iota 6503 df-fun 6553 df-fv 6559 df-1st 7997 df-2nd 7998

This theorem is referenced by: (None)

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