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Mirrors > Home > NFE Home > Th. List > funres | GIF version |
Description: A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 16-Aug-1994.) |
Ref | Expression |
---|---|
funres | ⊢ (Fun F → Fun (F ↾ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resss 4988 | . 2 ⊢ (F ↾ A) ⊆ F | |
2 | funss 5126 | . 2 ⊢ ((F ↾ A) ⊆ F → (Fun F → Fun (F ↾ A))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Fun F → Fun (F ↾ A)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3257 ↾ cres 4774 Fun wfun 4775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-opab 4623 df-br 4640 df-co 4726 df-cnv 4785 df-res 4788 df-fun 4789 |
This theorem is referenced by: fnssresb 5195 fnresi 5200 fores 5278 respreima 5410 funfvima 5459 |
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