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Mirrors > Home > ILE Home > Th. List > reseq1d | GIF version |
Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
reseqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
reseq1d | ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | reseq1 4900 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ↾ cres 4627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-res 4637 |
This theorem is referenced by: reseq12d 4907 fun2ssres 5258 funcnvres2 5290 funimaexg 5299 fresin 5393 offres 6133 tfrlemisucaccv 6323 tfrlemi1 6330 tfr1onlemsucaccv 6339 tfrcllemsucaccv 6352 freceq1 6390 freceq2 6391 fseq1p1m1 10089 setsresg 12492 setscom 12494 dvcoapbr 14042 bj-charfundcALT 14421 |
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