Theorem fnssresb 6664

Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)

Assertion

Ref	Expression
fnssresb	⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴))

Proof of Theorem fnssresb

Step	Hyp	Ref	Expression
1		df-fn 6544	. 2 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵))
2		fnfun 6642	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3	2	funresd 6586	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun (𝐹 ↾ 𝐵))
4	3	biantrurd 542	. . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵)))
5		ssdmres 6005	. . . 4 ⊢ (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐵) = 𝐵)
6		fndm 6645	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	sseq2d 3941	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
8	5, 7	bitr3id 288	. . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ 𝐵 ⊆ 𝐴))
9	4, 8	bitr3d 284	. 2 ⊢ (𝐹 Fn 𝐴 → ((Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵) ↔ 𝐵 ⊆ 𝐴))
10	1, 9	bitrid 286	1 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ⊆ wss 3896  dom cdm 5650   ↾ cres 5652  Fun wfun 6535   Fn wfn 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-fun 6543  df-fn 6544

This theorem is used by:  fnssres  6665  wrdred1hash  14711  rhmsscrnghm  20924  rngcrescrhm  20943  plyreres  26611  xrge0pluscn  34620  icoreresf  38382  fnbrafvb  48331  rngcrescrhmALTV  49484