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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 7096 | . . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1st2nd2 7096 | . . 3 ⊢ (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
3 | 1, 2 | eqeqan12d 2626 | . 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
4 | fvex 6113 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
5 | fvex 6113 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
6 | 4, 5 | opth 4871 | . 2 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))) |
7 | 3, 6 | syl6rbb 276 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: fseqdom 8732 iundom2g 9241 mdetunilem9 20245 txhaus 21260 fsumvma 24738 2wlkeq 26235 disjxpin 28783 poimirlem4 32583 poimirlem13 32592 poimirlem14 32593 poimirlem22 32601 poimirlem26 32605 poimirlem27 32606 rmxypairf1o 36494 1wlkeq 40838 |
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