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Mirrors > Home > ILE Home > Th. List > resmptf | GIF version |
Description: Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.) |
Ref | Expression |
---|---|
resmptf.a | ⊢ Ⅎ𝑥𝐴 |
resmptf.b | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
resmptf | ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resmpt 4867 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)) | |
2 | resmptf.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2281 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2281 | . . . 4 ⊢ Ⅎ𝑦𝐶 | |
5 | nfcsb1v 3035 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
6 | csbeq1a 3012 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
7 | 2, 3, 4, 5, 6 | cbvmptf 4022 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
8 | 7 | reseq1i 4815 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) |
9 | resmptf.b | . . 3 ⊢ Ⅎ𝑥𝐵 | |
10 | nfcv 2281 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
11 | 9, 10, 4, 5, 6 | cbvmptf 4022 | . 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
12 | 1, 8, 11 | 3eqtr4g 2197 | 1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Ⅎwnfc 2268 ⦋csb 3003 ⊆ wss 3071 ↦ cmpt 3989 ↾ cres 4541 |
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-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 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 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-opab 3990 df-mpt 3991 df-xp 4545 df-rel 4546 df-res 4551 |
This theorem is referenced by: (None) |
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