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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 4623 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V)) | |
2 | 1 | ineq2d 3328 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V))) |
3 | df-res 4621 | . 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V)) | |
4 | df-res 4621 | . 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)) | |
5 | 2, 3, 4 | 3eqtr4g 2228 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 Vcvv 2730 ∩ cin 3120 × cxp 4607 ↾ cres 4611 |
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-opab 4049 df-xp 4615 df-res 4621 |
This theorem is referenced by: reseq2i 4886 reseq2d 4889 resabs1 4918 resima2 4923 imaeq2 4947 resdisj 5037 relcoi1 5140 fressnfv 5681 tfrlem1 6285 tfrlem9 6296 tfr0dm 6299 tfrlemisucaccv 6302 tfrlemiubacc 6307 tfr1onlemsucaccv 6318 tfr1onlemubacc 6323 tfr1onlemaccex 6325 tfrcllemsucaccv 6331 tfrcllembxssdm 6333 tfrcllemubacc 6336 tfrcllemaccex 6338 tfrcllemres 6339 tfrcldm 6340 fnfi 6911 lmbr2 12969 lmff 13004 |
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