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Mirrors > Home > ILE Home > Th. List > eqopi | 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 4687 | . . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V) | |
2 | 1 | sseli 3120 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V)) |
3 | elxp6 6107 | . . . 4 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V))) | |
4 | 3 | simplbi 272 | . . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
5 | opeq12 3739 | . . 3 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈𝐵, 𝐶〉) | |
6 | 4, 5 | sylan9eq 2207 | . 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = 〈𝐵, 𝐶〉) |
7 | 2, 6 | sylan 281 | 1 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = 〈𝐵, 𝐶〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 2125 Vcvv 2709 〈cop 3559 × cxp 4577 ‘cfv 5163 1st c1st 6076 2nd c2nd 6077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 ax-un 4388 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ral 2437 df-rex 2438 df-v 2711 df-sbc 2934 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-br 3962 df-opab 4022 df-mpt 4023 df-id 4248 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-iota 5128 df-fun 5165 df-fv 5171 df-1st 6078 df-2nd 6079 |
This theorem is referenced by: op1steq 6117 dfoprab3 6129 1stconst 6158 2ndconst 6159 cnvoprab 6171 upxp 12611 |
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