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Mirrors > Home > ILE Home > Th. List > reseq1 | GIF version |
Description: Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
reseq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 3270 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ (𝐶 × V)) = (𝐵 ∩ (𝐶 × V))) | |
2 | df-res 4551 | . 2 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
3 | df-res 4551 | . 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
4 | 1, 2, 3 | 3eqtr4g 2197 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Vcvv 2686 ∩ cin 3070 × cxp 4537 ↾ 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-res 4551 |
This theorem is referenced by: reseq1i 4815 reseq1d 4818 imaeq1 4876 relcoi1 5070 tfr0dm 6219 tfrlemiex 6228 tfr1onlemex 6244 tfr1onlemaccex 6245 tfrcllemsucaccv 6251 tfrcllembxssdm 6253 tfrcllemex 6257 tfrcllemaccex 6258 tfrcllemres 6259 pmresg 6570 lmbr 12382 |
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