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Mirrors > Home > NFE Home > Th. List > reseq2 | GIF version |
Description: Equality theorem for restrictions. (Contributed by set.mm contributors, 8-Aug-1994.) |
Ref | Expression |
---|---|
reseq2 | ⊢ (A = B → (C ↾ A) = (C ↾ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1 4798 | . . 3 ⊢ (A = B → (A × V) = (B × V)) | |
2 | 1 | ineq2d 3457 | . 2 ⊢ (A = B → (C ∩ (A × V)) = (C ∩ (B × V))) |
3 | df-res 4788 | . 2 ⊢ (C ↾ A) = (C ∩ (A × V)) | |
4 | df-res 4788 | . 2 ⊢ (C ↾ B) = (C ∩ (B × V)) | |
5 | 2, 3, 4 | 3eqtr4g 2410 | 1 ⊢ (A = B → (C ↾ A) = (C ↾ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 Vcvv 2859 ∩ cin 3208 × cxp 4770 ↾ cres 4774 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-opab 4623 df-xp 4784 df-res 4788 |
This theorem is referenced by: reseq2i 4931 reseq2d 4934 resdisj 5050 fressnfv 5439 |
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