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Theorem xpopth 6177

Description: An ordered pair theorem for members of cross 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 6176	. . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2nd2 6176	. . 3 ⊢ (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	1, 2	eqeqan12d 2193	. 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
4		1stexg 6168	. . . 4 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → (1^st ‘𝐴) ∈ V)
5		2ndexg 6169	. . . 4 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → (2^nd ‘𝐴) ∈ V)
6		opthg 4239	. . . 4 ⊢ (((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V) → (⟨(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	4, 5, 6	syl2anc 411	. . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
8	7	adantr 276	. 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
9	3, 8	bitr2d 189	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 set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2738 ⟨cop 3596 × cxp 4625 ‘cfv 5217 1^st c1st 6139 2^nd c2nd 6140

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2740 df-sbc 2964 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fo 5223 df-fv 5225 df-1st 6141 df-2nd 6142

This theorem is referenced by: xmetxp 14010