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Mirrors > Home > MPE Home > Th. List > fnssresb | Structured version Visualization version GIF version |
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.) |
Ref | Expression |
---|---|
fnssresb | ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fn 6566 | . 2 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵)) | |
2 | fnfun 6669 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
3 | 2 | funresd 6611 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun (𝐹 ↾ 𝐵)) |
4 | 3 | biantrurd 532 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵))) |
5 | ssdmres 6033 | . . . 4 ⊢ (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐵) = 𝐵) | |
6 | fndm 6672 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
7 | 6 | sseq2d 4028 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
8 | 5, 7 | bitr3id 285 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
9 | 4, 8 | bitr3d 281 | . 2 ⊢ (𝐹 Fn 𝐴 → ((Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵) ↔ 𝐵 ⊆ 𝐴)) |
10 | 1, 9 | bitrid 283 | 1 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ⊆ wss 3963 dom cdm 5689 ↾ cres 5691 Fun wfun 6557 Fn wfn 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-fun 6565 df-fn 6566 |
This theorem is referenced by: fnssres 6692 wrdred1hash 14596 rhmsscrnghm 20682 rngcrescrhm 20701 plyreres 26339 xrge0pluscn 33901 icoreresf 37335 fnbrafvb 47104 rngcrescrhmALTV 48124 |
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