Theorem fnssresb 5370

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 5261	. 2 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵))
2		fnfun 5355	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3		funres 5299	. . . . 5 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐵))
4	2, 3	syl 14	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun (𝐹 ↾ 𝐵))
5	4	biantrurd 305	. . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵)))
6		ssdmres 4968	. . . 4 ⊢ (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐵) = 𝐵)
7		fndm 5357	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	7	sseq2d 3213	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
9	6, 8	bitr3id 194	. . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ 𝐵 ⊆ 𝐴))
10	5, 9	bitr3d 190	. 2 ⊢ (𝐹 Fn 𝐴 → ((Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵) ↔ 𝐵 ⊆ 𝐴))
11	1, 10	bitrid 192	1 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ⊆ wss 3157  dom cdm 4663   ↾ cres 4665  Fun wfun 5252   Fn wfn 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-res 4675  df-fun 5260  df-fn 5261

This theorem is referenced by:  fnssres  5371  wrdred1hash  10978  plyreres  15000