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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 7822 | . . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1st2nd2 7822 | . . 3 ⊢ (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
3 | 1, 2 | eqeqan12d 2753 | . 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
4 | fvex 6752 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
5 | fvex 6752 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
6 | 4, 5 | opth 5377 | . 2 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))) |
7 | 3, 6 | bitr2di 291 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 〈cop 4564 × cxp 5567 ‘cfv 6401 1st c1st 7781 2nd c2nd 7782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pr 5339 ax-un 7545 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5153 df-id 5472 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-iota 6359 df-fun 6403 df-fv 6409 df-1st 7783 df-2nd 7784 |
This theorem is referenced by: fseqdom 9670 iundom2g 10184 mdetunilem9 21549 txhaus 22576 fsumvma 26126 wlkeq 27753 disjxpin 30678 poimirlem4 35555 poimirlem13 35564 poimirlem14 35565 poimirlem22 35573 poimirlem26 35577 poimirlem27 35578 rmxypairf1o 40484 |
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