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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 7961 | . . 3 ⊢ (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) | |
2 | 1st2nd2 7961 | . . 3 ⊢ (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩) | |
3 | 1, 2 | eqeqan12d 2751 | . 2 ⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)) |
4 | fvex 6856 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
5 | fvex 6856 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
6 | 4, 5 | opth 5434 | . 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 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4593 × cxp 5632 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-1st 7922 df-2nd 7923 |
This theorem is referenced by: fseqdom 9963 iundom2g 10477 mdetunilem9 21972 txhaus 23001 fsumvma 26564 wlkeq 28585 disjxpin 31509 poimirlem4 36085 poimirlem13 36094 poimirlem14 36095 poimirlem22 36103 poimirlem26 36107 poimirlem27 36108 rmxypairf1o 41238 |
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