fv1stcnv - Mathbox for Scott Fenton

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Theorem fv1stcnv 35366

Description: The value of the converse of 1^st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)

**Assertion**
Ref	Expression
fv1stcnv	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (^◡(1^st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

**Proof of Theorem fv1stcnv**
Step	Hyp	Ref	Expression
1		snidg 4658	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ {𝑌})
2	1	anim2i 616	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
3		eqid 2728	. . . 4 ⊢ 𝑋 = 𝑋
4	2, 3	jctir 520	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋))
5		opex 5460	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
6		brcnvg 5876	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋))
7	5, 6	mpan2 690	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋))
8		brres 5986	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (⟨𝑋, 𝑌⟩(1^st ↾ (𝐴 × {𝑌}))𝑋 ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋)))
9	7, 8	bitrd 279	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋)))
10	9	adantr 480	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋)))
11		opelxp 5708	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}))
12	11	anbi1i 623	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋))
13		br1steqg 8009	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (⟨𝑋, 𝑌⟩1^st 𝑋 ↔ 𝑋 = 𝑋))
14	13	anbi2d 629	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
15	12, 14	bitrid 283	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1^st 𝑋) ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
16	10, 15	bitrd 279	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
17	4, 16	mpbird 257	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → 𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩)
18		1stconst 8099	. . . 4 ⊢ (𝑌 ∈ 𝑉 → (1^st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴)
19		f1ocnv 6845	. . . 4 ⊢ ((1^st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto→𝐴 → ^◡(1^st ↾ (𝐴 × {𝑌})):𝐴–1-1-onto→(𝐴 × {𝑌}))
20		f1ofn 6834	. . . 4 ⊢ (^◡(1^st ↾ (𝐴 × {𝑌})):𝐴–1-1-onto→(𝐴 × {𝑌}) → ^◡(1^st ↾ (𝐴 × {𝑌})) Fn 𝐴)
21	18, 19, 20	3syl 18	. . 3 ⊢ (𝑌 ∈ 𝑉 → ^◡(1^st ↾ (𝐴 × {𝑌})) Fn 𝐴)
22		simpl 482	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝐴)
23		fnbrfvb 6944	. . 3 ⊢ ((^◡(1^st ↾ (𝐴 × {𝑌})) Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((^◡(1^st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
24	21, 22, 23	syl2an2 685	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → ((^◡(1^st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋^◡(1^st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
25	17, 24	mpbird 257	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (^◡(1^st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 {csn 4624 ⟨cop 4630 class class class wbr 5142 × cxp 5670 ^◡ccnv 5671 ↾ cres 5674 Fn wfn 6537 –1-1-onto→wf1o 6541 ‘cfv 6542 1^st c1st 7985

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-1st 7987 df-2nd 7988

This theorem is referenced by: (None)

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