Theorem fnssresb 4954

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 4848	. 2 ⊢ ((𝐹 ↾ B) Fn B ↔ (Fun (𝐹 ↾ B) ∧ dom (𝐹 ↾ B) = B))
2		fnfun 4939	. . . . 5 ⊢ (𝐹 Fn A → Fun 𝐹)
3		funres 4884	. . . . 5 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ B))
4	2, 3	syl 14	. . . 4 ⊢ (𝐹 Fn A → Fun (𝐹 ↾ B))
5	4	biantrurd 289	. . 3 ⊢ (𝐹 Fn A → (dom (𝐹 ↾ B) = B ↔ (Fun (𝐹 ↾ B) ∧ dom (𝐹 ↾ B) = B)))
6		ssdmres 4576	. . . 4 ⊢ (B ⊆ dom 𝐹 ↔ dom (𝐹 ↾ B) = B)
7		fndm 4941	. . . . 5 ⊢ (𝐹 Fn A → dom 𝐹 = A)
8	7	sseq2d 2967	. . . 4 ⊢ (𝐹 Fn A → (B ⊆ dom 𝐹 ↔ B ⊆ A))
9	6, 8	syl5bbr 183	. . 3 ⊢ (𝐹 Fn A → (dom (𝐹 ↾ B) = B ↔ B ⊆ A))
10	5, 9	bitr3d 179	. 2 ⊢ (𝐹 Fn A → ((Fun (𝐹 ↾ B) ∧ dom (𝐹 ↾ B) = B) ↔ B ⊆ A))
11	1, 10	syl5bb 181	1 ⊢ (𝐹 Fn A → ((𝐹 ↾ B) Fn B ↔ B ⊆ A))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ⊆ wss 2911  dom cdm 4288   ↾ cres 4290  Fun wfun 4839   Fn wfn 4840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-fun 4847  df-fn 4848

This theorem is referenced by:  fnssres  4955