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Mirrors > Home > ILE Home > Th. List > reseq2i | GIF version |
Description: Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
reseqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
reseq2i | ⊢ (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | reseq2 4809 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ↾ cres 4536 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-opab 3985 df-xp 4540 df-res 4546 |
This theorem is referenced by: reseq12i 4812 rescom 4839 resdmdfsn 4857 rescnvcnv 4996 resdm2 5024 funcnvres 5191 funimaexg 5202 resdif 5382 frecfnom 6291 facnn 10466 fac0 10467 expcnv 11266 setsslid 11998 uptx 12432 txcn 12433 |
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