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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 5535 | . . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V) | |
2 | 1 | sseli 3911 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V)) |
3 | elxp6 7705 | . . . 4 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V))) | |
4 | 3 | simplbi 501 | . . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
5 | opeq12 4767 | . . 3 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉) | |
6 | 4, 5 | sylan9eq 2853 | . 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = 〈𝐵, 𝐶〉) |
7 | 2, 6 | sylan 583 | 1 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = 〈𝐵, 𝐶〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 × cxp 5517 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: op1steq 7715 el2xptp0 7718 dfoprab3 7734 1stconst 7778 2ndconst 7779 upxp 22228 opreu2reuALT 30247 cnvoprabOLD 30482 gsummpt2d 30734 sitgaddlemb 31716 |
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