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Theorem fnssresb 6457
 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 6343 . 2 ((𝐹𝐵) Fn 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵))
2 fnfun 6439 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
3 funres 6382 . . . . 5 (Fun 𝐹 → Fun (𝐹𝐵))
42, 3syl 17 . . . 4 (𝐹 Fn 𝐴 → Fun (𝐹𝐵))
54biantrurd 536 . . 3 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = 𝐵 ↔ (Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵)))
6 ssdmres 5851 . . . 4 (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹𝐵) = 𝐵)
7 fndm 6441 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
87sseq2d 3926 . . . 4 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
96, 8bitr3id 288 . . 3 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = 𝐵𝐵𝐴))
105, 9bitr3d 284 . 2 (𝐹 Fn 𝐴 → ((Fun (𝐹𝐵) ∧ dom (𝐹𝐵) = 𝐵) ↔ 𝐵𝐴))
111, 10syl5bb 286 1 (𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ⊆ wss 3860  dom cdm 5528   ↾ cres 5530  Fun wfun 6334   Fn wfn 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-fun 6342  df-fn 6343 This theorem is referenced by:  fnssres  6458  wrdred1hash  13973  plyreres  24992  xrge0pluscn  31424  icoreresf  35084  fnbrafvb  44137  rhmsscrnghm  45076  rngcrescrhm  45135  rngcrescrhmALTV  45153
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