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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 4511 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V)) | |
2 | 1 | ineq2d 3241 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V))) |
3 | df-res 4509 | . 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V)) | |
4 | df-res 4509 | . 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)) | |
5 | 2, 3, 4 | 3eqtr4g 2170 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 Vcvv 2655 ∩ cin 3034 × cxp 4495 ↾ cres 4499 |
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 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-v 2657 df-in 3041 df-opab 3948 df-xp 4503 df-res 4509 |
This theorem is referenced by: reseq2i 4772 reseq2d 4775 resabs1 4804 resima2 4809 imaeq2 4833 resdisj 4923 relcoi1 5026 fressnfv 5559 tfrlem1 6157 tfrlem9 6168 tfr0dm 6171 tfrlemisucaccv 6174 tfrlemiubacc 6179 tfr1onlemsucaccv 6190 tfr1onlemubacc 6195 tfr1onlemaccex 6197 tfrcllemsucaccv 6203 tfrcllembxssdm 6205 tfrcllemubacc 6208 tfrcllemaccex 6210 tfrcllemres 6211 tfrcldm 6212 fnfi 6775 lmbr2 12219 lmff 12254 |
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