Theorem pmresg 8845

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 4305	. . . . 5 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → ¬ (𝐴 ↑pm 𝐶) = ∅)
2		fnpm 8809	. . . . . . 7 ⊢ ↑pm Fn (V × V)
3	2	fndmi 6624	. . . . . 6 ⊢ dom ↑pm = (V × V)
4	3	ndmov 7575	. . . . 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 8821	. . . . . 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 4202	. . . 4 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
13		fssres 6728	. . . 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 6693	. . . . 5 ⊢ (𝐹:dom 𝐹⟶𝐴 → Fun 𝐹)
16		resres 5965	. . . . . 6 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
17		funrel 6535	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
18		resdm 5999	. . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
19		reseq1 5946	. . . . . . 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 2779	. . . . 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 6672	. . 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 4203	. . 3 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ 𝐵
26		elpm2r 8820	. . 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 1540   ∈ wcel 2109  Vcvv 3450   ∩ cin 3915   ⊆ wss 3916  ∅c0 4298   × cxp 5638  dom cdm 5640   ↾ cres 5642  Rel wrel 5645  Fun wfun 6507  ⟶wf 6509  (class class class)co 7389   ↑_pm cpm 8802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-pm 8804

This theorem is referenced by:  lmres  23193  mbfres  25551  dvnres  25839  cpnres  25845  caures  37749