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Theorem xpopth 8012

Description: An ordered pair theorem for members of Cartesian products. (Contributed by NM, 20-Jun-2007.)

**Assertion**
Ref	Expression
xpopth	⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ 𝐴 = 𝐵))

**Proof of Theorem xpopth**
Step	Hyp	Ref	Expression
1		1st2nd2 8010	. . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2nd2 8010	. . 3 ⊢ (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	1, 2	eqeqan12d 2746	. 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
4		fvex 6901	. . 3 ⊢ (1^st ‘𝐴) ∈ V
5		fvex 6901	. . 3 ⊢ (2^nd ‘𝐴) ∈ V
6	4, 5	opth 5475	. 2 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)))
7	3, 6	bitr2di 287	1 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ 𝐴 = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⟨cop 4633 × cxp 5673 ‘cfv 6540 1^st c1st 7969 2^nd c2nd 7970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fv 6548 df-1st 7971 df-2nd 7972

This theorem is referenced by: fseqdom 10017 iundom2g 10531 mdetunilem9 22113 txhaus 23142 fsumvma 26705 wlkeq 28880 disjxpin 31806 poimirlem4 36480 poimirlem13 36489 poimirlem14 36490 poimirlem22 36498 poimirlem26 36502 poimirlem27 36503 rmxypairf1o 41635