Theorem pmresg 8902

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 4285	. . . . 5 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → ¬ (𝐴 ↑pm 𝐶) = ∅)
2		fnpm 8858	. . . . . . 7 ⊢ ↑pm Fn (V × V)
3	2	fndmi 6644	. . . . . 6 ⊢ dom ↑pm = (V × V)
4	3	ndmov 7606	. . . . 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 500	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → 𝐴 ∈ V)
7	6	adantl 487	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐴 ∈ V)
8		simpl 488	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐵 ∈ 𝑉)
9		elpmi 8870	. . . . . 6 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐶))
10	9	simpld 500	. . . . 5 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → 𝐹:dom 𝐹⟶𝐴)
11	10	adantl 487	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐹:dom 𝐹⟶𝐴)
12		inss1 4133	. . . 4 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
13		fssres 6749	. . . 4 ⊢ ((𝐹:dom 𝐹⟶𝐴 ∧ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴)
14	11, 12, 13	sylancl 598	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴)
15		ffun 6713	. . . . 5 ⊢ (𝐹:dom 𝐹⟶𝐴 → Fun 𝐹)
16		resres 5982	. . . . . 6 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
17		funrel 6557	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
18		resdm 6014	. . . . . . 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 2809	. . . . 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 6692	. . 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 4134	. . 3 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ 𝐵
26		elpm2r 8869	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ ((𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴 ∧ (dom 𝐹 ∩ 𝐵) ⊆ 𝐵)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))
27	25, 26	mpanr2 717	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))
28	7, 8, 24, 27	syl21anc 851	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278   × cxp 5648  dom cdm 5650   ↾ cres 5652  Rel wrel 5655  Fun wfun 6533  ⟶wf 6535  (class class class)co 7421   ↑_pm cpm 8852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-pm 8854

This theorem is used by:  lmres  23624  mbfres  25971  dvnres  26257  cpnres  26263  caures  38788