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Mirrors > Home > ILE Home > Th. List > csbopeq1a | GIF version |
Description: Equality theorem for substitution of a class 𝐴 for an ordered pair 〈𝑥, 𝑦〉 in 𝐵 (analog of csbeq1a 3012). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
csbopeq1a | ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | vex 2689 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | op2ndd 6047 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = 𝑦) |
4 | 3 | eqcomd 2145 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑦 = (2nd ‘𝐴)) |
5 | csbeq1a 3012 | . . 3 ⊢ (𝑦 = (2nd ‘𝐴) → 𝐵 = ⦋(2nd ‘𝐴) / 𝑦⦌𝐵) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝐵 = ⦋(2nd ‘𝐴) / 𝑦⦌𝐵) |
7 | 1, 2 | op1std 6046 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = 𝑥) |
8 | 7 | eqcomd 2145 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑥 = (1st ‘𝐴)) |
9 | csbeq1a 3012 | . . 3 ⊢ (𝑥 = (1st ‘𝐴) → ⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵) | |
10 | 8, 9 | syl 14 | . 2 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵) |
11 | 6, 10 | eqtr2d 2173 | 1 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ⦋csb 3003 〈cop 3530 ‘cfv 5123 1st c1st 6036 2nd c2nd 6037 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: dfmpo 6120 f1od2 6132 |
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