Theorem fnssresb 6469

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 6358	. 2 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵))
2		fnfun 6453	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3		funres 6397	. . . . 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 5876	. . . 4 ⊢ (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐵) = 𝐵)
7		fndm 6455	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	7	sseq2d 3999	. . . 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))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ⊆ wss 3936  dom cdm 5555   ↾ cres 5557  Fun wfun 6349   Fn wfn 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-res 5567  df-fun 6357  df-fn 6358

This theorem is referenced by:  fnssres  6470  wrdred1hash  13913  plyreres  24872  xrge0pluscn  31183  icoreresf  34636  fnbrafvb  43373  rhmsscrnghm  44317  rngcrescrhm  44376  rngcrescrhmALTV  44394