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Mirrors > Home > MPE Home > Th. List > eqopi | Structured version Visualization version GIF version |
Description: Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
eqopi | ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = 〈𝐵, 𝐶〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 5705 | . . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V) | |
2 | 1 | sseli 3991 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V)) |
3 | elxp6 8047 | . . . 4 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V))) | |
4 | 3 | simplbi 497 | . . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
5 | opeq12 4880 | . . 3 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉) | |
6 | 4, 5 | sylan9eq 2795 | . 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = 〈𝐵, 𝐶〉) |
7 | 2, 6 | sylan 580 | 1 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = 〈𝐵, 𝐶〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 × cxp 5687 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: op1steq 8057 el2xptp0 8060 dfoprab3 8078 1stconst 8124 2ndconst 8125 upxp 23647 opreu2reuALT 32505 cnvoprabOLD 32738 gsummpt2d 33035 sitgaddlemb 34330 |
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