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Theorem xpopth 7446

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 7444	. . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2nd2 7444	. . 3 ⊢ (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	1, 2	eqeqan12d 2819	. 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
4		fvex 6428	. . 3 ⊢ (1^st ‘𝐴) ∈ V
5		fvex 6428	. . 3 ⊢ (2^nd ‘𝐴) ∈ V
6	4, 5	opth 5139	. 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	syl6rbb 280	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 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ⟨cop 4378 × cxp 5314 ‘cfv 6105 1^st c1st 7403 2^nd c2nd 7404

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2379 ax-ext 2781 ax-sep 4979 ax-nul 4987 ax-pow 5039 ax-pr 5101 ax-un 7187

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2593 df-eu 2611 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ral 3098 df-rex 3099 df-rab 3102 df-v 3391 df-sbc 3638 df-dif 3776 df-un 3778 df-in 3780 df-ss 3787 df-nul 4120 df-if 4282 df-sn 4373 df-pr 4375 df-op 4379 df-uni 4633 df-br 4848 df-opab 4910 df-mpt 4927 df-id 5224 df-xp 5322 df-rel 5323 df-cnv 5324 df-co 5325 df-dm 5326 df-rn 5327 df-iota 6068 df-fun 6107 df-fv 6113 df-1st 7405 df-2nd 7406

This theorem is referenced by: fseqdom 9139 iundom2g 9654 mdetunilem9 20756 txhaus 21783 fsumvma 25294 wlkeq 26887 disjxpin 29922 poimirlem4 33906 poimirlem13 33915 poimirlem14 33916 poimirlem22 33924 poimirlem26 33928 poimirlem27 33929 rmxypairf1o 38265