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Theorem xpopth 6348

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 6347	. . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2nd2 6347	. . 3 ⊢ (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	1, 2	eqeqan12d 2247	. 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
4		1stexg 6339	. . . 4 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → (1^st ‘𝐴) ∈ V)
5		2ndexg 6340	. . . 4 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → (2^nd ‘𝐴) ∈ V)
6		opthg 4336	. . . 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 1398 ∈ wcel 2202 Vcvv 2803 ⟨cop 3676 × cxp 4729 ‘cfv 5333 1^st c1st 6310 2^nd c2nd 6311

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-v 2805 df-sbc 3033 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-fo 5339 df-fv 5341 df-1st 6312 df-2nd 6313

This theorem is referenced by: xmetxp 15318 wlkeq 16295