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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 6545 | . 2 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵)) | |
2 | fnfun 6648 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
3 | 2 | funresd 6590 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun (𝐹 ↾ 𝐵)) |
4 | 3 | biantrurd 531 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵))) |
5 | ssdmres 6003 | . . . 4 ⊢ (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐵) = 𝐵) | |
6 | fndm 6651 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
7 | 6 | sseq2d 4013 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
8 | 5, 7 | bitr3id 284 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
9 | 4, 8 | bitr3d 280 | . 2 ⊢ (𝐹 Fn 𝐴 → ((Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵) ↔ 𝐵 ⊆ 𝐴)) |
10 | 1, 9 | bitrid 282 | 1 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ⊆ wss 3947 dom cdm 5675 ↾ cres 5677 Fun wfun 6536 Fn wfn 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-res 5687 df-fun 6544 df-fn 6545 |
This theorem is referenced by: fnssres 6672 wrdred1hash 14515 plyreres 26032 xrge0pluscn 33218 icoreresf 36536 fnbrafvb 46160 rhmsscrnghm 47012 rngcrescrhm 47071 rngcrescrhmALTV 47089 |
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