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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 4771 | . 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 1314 ↾ cres 4501 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-in 3043 df-res 4511 |
This theorem is referenced by: reseq12d 4778 fun2ssres 5124 funcnvres2 5156 funimaexg 5165 fresin 5259 offres 5987 tfrlemisucaccv 6176 tfrlemi1 6183 tfr1onlemsucaccv 6192 tfrcllemsucaccv 6205 freceq1 6243 freceq2 6244 fseq1p1m1 9764 setsresg 11837 setscom 11839 |
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