eqopi - Metamath Proof Explorer

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Theorem eqopi 8023

Description: Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)

**Assertion**
Ref	Expression
eqopi	⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1^st ‘𝐴) = 𝐵 ∧ (2^nd ‘𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)

**Proof of Theorem eqopi**
Step	Hyp	Ref	Expression
1		xpss 5689	. . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
2	1	sseli 3969	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3		elxp6 8021	. . . 4 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)))
4	3	simplbi 496	. . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
5		opeq12 4872	. . 3 ⊢ (((1^st ‘𝐴) = 𝐵 ∧ (2^nd ‘𝐴) = 𝐶) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩)
6	4, 5	sylan9eq 2785	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) = 𝐵 ∧ (2^nd ‘𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
7	2, 6	sylan 578	1 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1^st ‘𝐴) = 𝐵 ∧ (2^nd ‘𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⟨cop 4631 × cxp 5671 ‘cfv 6543 1^st c1st 7985 2^nd c2nd 7986

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 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7735

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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7987 df-2nd 7988

This theorem is referenced by: op1steq 8031 el2xptp0 8034 dfoprab3 8052 1stconst 8098 2ndconst 8099 upxp 23540 opreu2reuALT 32315 cnvoprabOLD 32542 gsummpt2d 32803 sitgaddlemb 34021

	
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