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Mirrors > Home > MPE Home > Th. List > xpopth | Structured version Visualization version GIF version |
Description: An ordered pair theorem for members of Cartesian products. (Contributed by NM, 20-Jun-2007.) |
Ref | Expression |
---|---|
xpopth | ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 8010 | . . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) | |
2 | 1st2nd2 8010 | . . 3 ⊢ (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩) | |
3 | 1, 2 | eqeqan12d 2740 | . 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)) |
4 | fvex 6897 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
5 | fvex 6897 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
6 | 4, 5 | opth 5469 | . 2 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))) |
7 | 3, 6 | bitr2di 288 | 1 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⟨cop 4629 × cxp 5667 ‘cfv 6536 1st c1st 7969 2nd c2nd 7970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fv 6544 df-1st 7971 df-2nd 7972 |
This theorem is referenced by: fseqdom 10020 iundom2g 10534 mdetunilem9 22473 txhaus 23502 fsumvma 27097 wlkeq 29396 disjxpin 32324 poimirlem4 37003 poimirlem13 37012 poimirlem14 37013 poimirlem22 37021 poimirlem26 37025 poimirlem27 37026 rmxypairf1o 42209 |
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