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Theorem fnssresb 6164
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
Assertion
Ref Expression
fnssresb (𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))

Proof of Theorem fnssresb
StepHypRef Expression
1 df-fn 6052 . 2 ((𝐹𝐵) Fn 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵))
2 fnfun 6149 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
3 funres 6090 . . . . 5 (Fun 𝐹 → Fun (𝐹𝐵))
42, 3syl 17 . . . 4 (𝐹 Fn 𝐴 → Fun (𝐹𝐵))
54biantrurd 530 . . 3 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵)))
6 ssdmres 5578 . . . 4 (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹𝐵) = 𝐵)
7 fndm 6151 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
87sseq2d 3774 . . . 4 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
96, 8syl5bbr 274 . . 3 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = 𝐵𝐵𝐴))
105, 9bitr3d 270 . 2 (𝐹 Fn 𝐴 → ((Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵) ↔ 𝐵𝐴))
111, 10syl5bb 272 1 (𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wss 3715  dom cdm 5266  cres 5268  Fun wfun 6043   Fn wfn 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-fun 6051  df-fn 6052
This theorem is referenced by:  fnssres  6165  wrdred1hash  13557  plyreres  24257  xrge0pluscn  30316  icoreresf  33529  fnbrafvb  41758  rhmsscrnghm  42554  rngcrescrhm  42613  rngcrescrhmALTV  42631
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