Theorem pmresg 8794

Description: Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Assertion

Ref	Expression
pmresg	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))

Proof of Theorem pmresg

Step	Hyp	Ref	Expression
1		n0i 4290	. . . . 5 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → ¬ (𝐴 ↑pm 𝐶) = ∅)
2		fnpm 8758	. . . . . . 7 ⊢ ↑pm Fn (V × V)
3	2	fndmi 6585	. . . . . 6 ⊢ dom ↑pm = (V × V)
4	3	ndmov 7530	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ↑pm 𝐶) = ∅)
5	1, 4	nsyl2 141	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
6	5	simpld 494	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → 𝐴 ∈ V)
7	6	adantl 481	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐴 ∈ V)
8		simpl 482	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐵 ∈ 𝑉)
9		elpmi 8770	. . . . . 6 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐶))
10	9	simpld 494	. . . . 5 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → 𝐹:dom 𝐹⟶𝐴)
11	10	adantl 481	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐹:dom 𝐹⟶𝐴)
12		inss1 4187	. . . 4 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
13		fssres 6689	. . . 4 ⊢ ((𝐹:dom 𝐹⟶𝐴 ∧ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴)
14	11, 12, 13	sylancl 586	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴)
15		ffun 6654	. . . . 5 ⊢ (𝐹:dom 𝐹⟶𝐴 → Fun 𝐹)
16		resres 5941	. . . . . 6 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
17		funrel 6498	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
18		resdm 5975	. . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
19		reseq1 5922	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) = 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
20	17, 18, 19	3syl 18	. . . . . 6 ⊢ (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
21	16, 20	eqtr3id 2780	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
22	11, 15, 21	3syl 18	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
23	22	feq1d 6633	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → ((𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴 ↔ (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴))
24	14, 23	mpbid 232	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴)
25		inss2 4188	. . 3 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ 𝐵
26		elpm2r 8769	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ ((𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴 ∧ (dom 𝐹 ∩ 𝐵) ⊆ 𝐵)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))
27	25, 26	mpanr2 704	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))
28	7, 8, 24, 27	syl21anc 837	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283   × cxp 5614  dom cdm 5616   ↾ cres 5618  Rel wrel 5621  Fun wfun 6475  ⟶wf 6477  (class class class)co 7346   ↑_pm cpm 8751

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-pm 8753

This theorem is referenced by:  lmres  23213  mbfres  25570  dvnres  25858  cpnres  25864  caures  37799