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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 6351 | . 2 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵)) | |
2 | fnfun 6446 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
3 | funres 6390 | . . . . 5 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐵)) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun (𝐹 ↾ 𝐵)) |
5 | 4 | biantrurd 535 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵))) |
6 | ssdmres 5869 | . . . 4 ⊢ (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐵) = 𝐵) | |
7 | fndm 6448 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
8 | 7 | sseq2d 3997 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
9 | 6, 8 | syl5bbr 287 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
10 | 5, 9 | bitr3d 283 | . 2 ⊢ (𝐹 Fn 𝐴 → ((Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵) ↔ 𝐵 ⊆ 𝐴)) |
11 | 1, 10 | syl5bb 285 | 1 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ⊆ wss 3934 dom cdm 5548 ↾ cres 5550 Fun wfun 6342 Fn wfn 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-fun 6350 df-fn 6351 |
This theorem is referenced by: fnssres 6463 wrdred1hash 13905 plyreres 24864 xrge0pluscn 31176 icoreresf 34625 fnbrafvb 43344 rhmsscrnghm 44288 rngcrescrhm 44347 rngcrescrhmALTV 44365 |
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