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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 3016). (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 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | vex 2692 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | op2ndd 6055 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = 𝑦) |
4 | 3 | eqcomd 2146 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑦 = (2nd ‘𝐴)) |
5 | csbeq1a 3016 | . . 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 6054 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = 𝑥) |
8 | 7 | eqcomd 2146 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑥 = (1st ‘𝐴)) |
9 | csbeq1a 3016 | . . 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 2174 | 1 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ⦋csb 3007 〈cop 3535 ‘cfv 5131 1st c1st 6044 2nd c2nd 6045 |
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 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fv 5139 df-1st 6046 df-2nd 6047 |
This theorem is referenced by: dfmpo 6128 f1od2 6140 |
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