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Mirrors > Home > ILE Home > Th. List > reseq2 | GIF version |
Description: Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.) |
Ref | Expression |
---|---|
reseq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1 4553 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V)) | |
2 | 1 | ineq2d 3277 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V))) |
3 | df-res 4551 | . 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V)) | |
4 | df-res 4551 | . 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)) | |
5 | 2, 3, 4 | 3eqtr4g 2197 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) |
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-opab 3990 df-xp 4545 df-res 4551 |
This theorem is referenced by: reseq2i 4816 reseq2d 4819 resabs1 4848 resima2 4853 imaeq2 4877 resdisj 4967 relcoi1 5070 fressnfv 5607 tfrlem1 6205 tfrlem9 6216 tfr0dm 6219 tfrlemisucaccv 6222 tfrlemiubacc 6227 tfr1onlemsucaccv 6238 tfr1onlemubacc 6243 tfr1onlemaccex 6245 tfrcllemsucaccv 6251 tfrcllembxssdm 6253 tfrcllemubacc 6256 tfrcllemaccex 6258 tfrcllemres 6259 tfrcldm 6260 fnfi 6825 lmbr2 12383 lmff 12418 |
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