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Theorem xpopth 5725

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 5724	. . 3 ⊢ (A ∈ (𝐶 × 𝐷) → A = ⟨(1^st ‘A), (2^nd ‘A)⟩)
2		1st2nd2 5724	. . 3 ⊢ (B ∈ (𝑅 × 𝑆) → B = ⟨(1^st ‘B), (2^nd ‘B)⟩)
3	1, 2	eqeqan12d 2037	. 2 ⊢ ((A ∈ (𝐶 × 𝐷) ∧ B ∈ (𝑅 × 𝑆)) → (A = B ↔ ⟨(1^st ‘A), (2^nd ‘A)⟩ = ⟨(1^st ‘B), (2^nd ‘B)⟩))
4		1stexg 5717	. . . 4 ⊢ (A ∈ (𝐶 × 𝐷) → (1^st ‘A) ∈ V)
5		2ndexg 5718	. . . 4 ⊢ (A ∈ (𝐶 × 𝐷) → (2^nd ‘A) ∈ V)
6		opthg 3949	. . . 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 393	. . 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 261	. 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 178	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 97 ↔ wb 98 = wceq 1228 ∈ wcel 1374 Vcvv 2535 ⟨cop 3353 × cxp 4270 ‘cfv 4829 1^st c1st 5688 2^nd c2nd 5689

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 617 ax-5 1316 ax-7 1317 ax-gen 1318 ax-ie1 1363 ax-ie2 1364 ax-8 1376 ax-10 1377 ax-11 1378 ax-i12 1379 ax-bnd 1380 ax-4 1381 ax-13 1385 ax-14 1386 ax-17 1400 ax-i9 1404 ax-ial 1409 ax-i5r 1410 ax-ext 2004 ax-sep 3849 ax-pow 3901 ax-pr 3918 ax-un 4120

This theorem depends on definitions: df-bi 110 df-3an 875 df-tru 1231 df-nf 1330 df-sb 1628 df-eu 1885 df-mo 1886 df-clab 2009 df-cleq 2015 df-clel 2018 df-nfc 2149 df-ral 2289 df-rex 2290 df-v 2537 df-sbc 2742 df-un 2899 df-in 2901 df-ss 2908 df-pw 3336 df-sn 3356 df-pr 3357 df-op 3359 df-uni 3555 df-br 3739 df-opab 3793 df-mpt 3794 df-id 4004 df-xp 4278 df-rel 4279 df-cnv 4280 df-co 4281 df-dm 4282 df-rn 4283 df-iota 4794 df-fun 4831 df-fn 4832 df-f 4833 df-fo 4835 df-fv 4837 df-1st 5690 df-2nd 5691

This theorem is referenced by: (None)