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Mirrors > Home > ILE Home > Th. List > reseq2d | GIF version |
Description: Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
reseqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
reseq2d | ⊢ (𝜑 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | reseq2 4814 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ↾ 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-in 3077 df-opab 3990 df-xp 4545 df-res 4551 |
This theorem is referenced by: reseq12d 4820 resima2 4853 relresfld 5068 f1orescnv 5383 funcocnv2 5392 fococnv2 5393 fnressn 5606 oprssov 5912 dftpos2 6158 fnsnsplitdc 6401 dif1en 6773 sbthlemi4 6848 fseq1p1m1 9874 resunimafz0 10574 setsvala 11990 metreslem 12549 xmspropd 12646 mspropd 12647 |
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