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Mirrors > Home > MPE Home > Th. List > eqop | Structured version Visualization version GIF version |
Description: Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
eqop | ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 8034 | . . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1 | eqeq1d 2728 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉)) |
3 | fvex 6906 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
4 | fvex 6906 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
5 | 3, 4 | opth 5474 | . 2 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) |
6 | 2, 5 | bitrdi 286 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 〈cop 4629 × cxp 5672 ‘cfv 6546 1st c1st 7993 2nd c2nd 7994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-iota 6498 df-fun 6548 df-fv 6554 df-1st 7995 df-2nd 7996 |
This theorem is referenced by: eqop2 8038 op1steq 8039 el2xptp0 8042 lsmhash 19699 txhmeo 23795 ptuncnv 23799 wlkcomp 29565 clwlkcomp 29713 f1od2 32635 esum2dlem 33938 poimirlem22 37356 rngosn3 37638 dvhb1dimN 40698 f1o2d2 41979 |
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