Theorem resfunexg 6479

Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)

Assertion

Ref	Expression
resfunexg	⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)

Proof of Theorem resfunexg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funres 5929	. . . . . . 7 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵))
2	1	adantr 481	. . . . . 6 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → Fun (𝐴 ↾ 𝐵))
3		funfn 5918	. . . . . 6 ⊢ (Fun (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
4	2, 3	sylib 208	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
5		dffn5 6241	. . . . 5 ⊢ ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)))
6	4, 5	sylib 208	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)))
7		fvex 6201	. . . . 5 ⊢ ((𝐴 ↾ 𝐵)‘𝑥) ∈ V
8	7	fnasrn 6411	. . . 4 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
9	6, 8	syl6eq 2672	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩))
10		opex 4932	. . . . . 6 ⊢ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩ ∈ V
11		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
12	10, 11	dmmpti 6023	. . . . 5 ⊢ dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) = dom (𝐴 ↾ 𝐵)
13	12	imaeq2i 5464	. . . 4 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)) = ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵))
14		imadmrn 5476	. . . 4 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
15	13, 14	eqtr3i 2646	. . 3 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
16	9, 15	syl6eqr 2674	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)))
17		funmpt 5926	. . 3 ⊢ Fun (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
18		dmresexg 5421	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V)
19	18	adantl 482	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V)
20		funimaexg 5975	. . 3 ⊢ ((Fun (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) ∧ dom (𝐴 ↾ 𝐵) ∈ V) → ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)) ∈ V)
21	17, 19, 20	sylancr 695	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)) ∈ V)
22	16, 21	eqeltrd 2701	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  resiexd  6480  fnex  6481  ofexg  6901  cofunexg  7130  dfac8alem  8852  dfac12lem1  8965  cfsmolem  9092  alephsing  9098  itunifval  9238  zorn2lem1  9318  ttukeylem3  9333  imadomg  9356  wunex2  9560  inar1  9597  axdc4uzlem  12782  hashf1rn  13142  hashf1rnOLD  13143  bpolylem  14779  1stf1  16832  1stf2  16833  2ndf1  16835  2ndf2  16836  1stfcl  16837  2ndfcl  16838  gsumzadd  18322  wlkreslem  26566  madeval  31935  tendo02  36075  dnnumch1  37614  aomclem6  37629  dfrngc2  41972  dfringc2  42018  rngcresringcat  42030  fdivval  42333