Theorem pmresg 8856

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 4295	. . . . 5 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → ¬ (𝐴 ↑pm 𝐶) = ∅)
2		fnpm 8819	. . . . . . 7 ⊢ ↑pm Fn (V × V)
3	2	fndmi 6629	. . . . . 6 ⊢ dom ↑pm = (V × V)
4	3	ndmov 7584	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ↑pm 𝐶) = ∅)
5	1, 4	nsyl2 142	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
6	5	simpld 499	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → 𝐴 ∈ V)
7	6	adantl 486	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐴 ∈ V)
8		simpl 487	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐵 ∈ 𝑉)
9		elpmi 8831	. . . . . 6 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐶))
10	9	simpld 499	. . . . 5 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → 𝐹:dom 𝐹⟶𝐴)
11	10	adantl 486	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐹:dom 𝐹⟶𝐴)
12		inss1 4191	. . . 4 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
13		fssres 6734	. . . 4 ⊢ ((𝐹:dom 𝐹⟶𝐴 ∧ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴)
14	11, 12, 13	sylancl 597	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴)
15		ffun 6698	. . . . 5 ⊢ (𝐹:dom 𝐹⟶𝐴 → Fun 𝐹)
16		resres 5982	. . . . . 6 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
17		funrel 6542	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
18		resdm 6016	. . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
19		reseq1 5963	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) = 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
20	17, 18, 19	3syl 19	. . . . . 6 ⊢ (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
21	16, 20	eqtr3id 2814	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
22	11, 15, 21	3syl 19	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
23	22	feq1d 6677	. . 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 235	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴)
25		inss2 4192	. . 3 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ 𝐵
26		elpm2r 8830	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ ((𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴 ∧ (dom 𝐹 ∩ 𝐵) ⊆ 𝐵)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))
27	25, 26	mpanr2 716	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))
28	7, 8, 24, 27	syl21anc 850	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288   × cxp 5650  dom cdm 5652   ↾ cres 5654  Rel wrel 5657  Fun wfun 6519  ⟶wf 6521  (class class class)co 7400   ↑_pm cpm 8813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-pm 8815

This theorem is referenced by:  lmres  23418  mbfres  25764  dvnres  26051  cpnres  26057  caures  38271