Theorem pmresg 6642

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

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pm 6617	. . . 4 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
2	1	elmpocl1 6037	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → 𝐴 ∈ V)
3	2	adantl 275	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐴 ∈ V)
4		simpl 108	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐵 ∈ 𝑉)
5		elpmi 6633	. . . . . 6 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐶))
6	5	simpld 111	. . . . 5 ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐶) → 𝐹:dom 𝐹⟶𝐴)
7	6	adantl 275	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → 𝐹:dom 𝐹⟶𝐴)
8		inss1 3342	. . . 4 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
9		fssres 5363	. . . 4 ⊢ ((𝐹:dom 𝐹⟶𝐴 ∧ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴)
10	7, 8, 9	sylancl 410	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴)
11		ffun 5340	. . . . 5 ⊢ (𝐹:dom 𝐹⟶𝐴 → Fun 𝐹)
12		resres 4896	. . . . . 6 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
13		funrel 5205	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
14		resdm 4923	. . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
15		reseq1 4878	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) = 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
16	13, 14, 15	3syl 17	. . . . . 6 ⊢ (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
17	12, 16	eqtr3id 2213	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
18	7, 11, 17	3syl 17	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
19	18	feq1d 5324	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → ((𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶𝐴 ↔ (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴))
20	10, 19	mpbid 146	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴)
21		inss2 3343	. . 3 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ 𝐵
22		elpm2r 6632	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ ((𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴 ∧ (dom 𝐹 ∩ 𝐵) ⊆ 𝐵)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))
23	21, 22	mpanr2 435	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶𝐴) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))
24	3, 4, 20, 23	syl21anc 1227	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  {crab 2448  Vcvv 2726   ∩ cin 3115   ⊆ wss 3116  𝒫 cpw 3559   × cxp 4602  dom cdm 4604   ↾ cres 4606  Rel wrel 4609  Fun wfun 5182  ⟶wf 5184  (class class class)co 5842   ↑_pm cpm 6615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pm 6617

This theorem is referenced by:  lmres  12888