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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 4359 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V)) | |
2 | 1 | ineq2d 3138 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V))) |
3 | df-res 4357 | . 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V)) | |
4 | df-res 4357 | . 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)) | |
5 | 2, 3, 4 | 3eqtr4g 2097 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 Vcvv 2557 ∩ cin 2916 × cxp 4343 ↾ cres 4347 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-opab 3819 df-xp 4351 df-res 4357 |
This theorem is referenced by: reseq2i 4609 reseq2d 4612 resabs1 4640 resima2 4644 imaeq2 4664 resdisj 4751 relcoi1 4849 fressnfv 5350 tfrlem1 5923 tfrlem9 5935 tfr0 5937 tfrlemisucaccv 5939 tfrlemiubacc 5944 |
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