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Mirrors > Home > NFE Home > Th. List > ssres | GIF version |
Description: Subclass theorem for restriction. (Contributed by set.mm contributors, 16-Aug-1994.) |
Ref | Expression |
---|---|
ssres | ⊢ (A ⊆ B → (A ↾ C) ⊆ (B ↾ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrin 3480 | . 2 ⊢ (A ⊆ B → (A ∩ (C × V)) ⊆ (B ∩ (C × V))) | |
2 | df-res 4788 | . 2 ⊢ (A ↾ C) = (A ∩ (C × V)) | |
3 | df-res 4788 | . 2 ⊢ (B ↾ C) = (B ∩ (C × V)) | |
4 | 1, 2, 3 | 3sstr4g 3312 | 1 ⊢ (A ⊆ B → (A ↾ C) ⊆ (B ↾ C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 2859 ∩ cin 3208 ⊆ wss 3257 × 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-ss 3259 df-res 4788 |
This theorem is referenced by: imass1 5023 |
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