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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 7722 | . . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1st2nd2 7722 | . . 3 ⊢ (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
3 | 1, 2 | eqeqan12d 2838 | . 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
4 | fvex 6677 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
5 | fvex 6677 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
6 | 4, 5 | opth 5360 | . 2 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))) |
7 | 3, 6 | syl6rbb 290 | 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4566 × cxp 5547 ‘cfv 6349 1st c1st 7681 2nd c2nd 7682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-1st 7683 df-2nd 7684 |
This theorem is referenced by: fseqdom 9446 iundom2g 9956 mdetunilem9 21223 txhaus 22249 fsumvma 25783 wlkeq 27409 disjxpin 30332 poimirlem4 34890 poimirlem13 34899 poimirlem14 34900 poimirlem22 34908 poimirlem26 34912 poimirlem27 34913 rmxypairf1o 39501 |
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