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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 4885 | . 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 1348 ↾ cres 4613 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-res 4623 |
This theorem is referenced by: reseq12d 4892 fun2ssres 5241 funcnvres2 5273 funimaexg 5282 fresin 5376 offres 6114 tfrlemisucaccv 6304 tfrlemi1 6311 tfr1onlemsucaccv 6320 tfrcllemsucaccv 6333 freceq1 6371 freceq2 6372 fseq1p1m1 10050 setsresg 12454 setscom 12456 dvcoapbr 13465 bj-charfundcALT 13844 |
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