Theorem resfunexg 6981

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 6400	. . . . . . 7 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵))
2	1	adantr 483	. . . . . 6 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → Fun (𝐴 ↾ 𝐵))
3	2	funfnd 6389	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
4		dffn5 6727	. . . . 5 ⊢ ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)))
5	3, 4	sylib 220	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)))
6		fvex 6686	. . . . 5 ⊢ ((𝐴 ↾ 𝐵)‘𝑥) ∈ V
7	6	fnasrn 6910	. . . 4 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
8	5, 7	syl6eq 2875	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩))
9		opex 5359	. . . . . 6 ⊢ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩ ∈ V
10		eqid 2824	. . . . . 6 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
11	9, 10	dmmpti 6495	. . . . 5 ⊢ dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) = dom (𝐴 ↾ 𝐵)
12	11	imaeq2i 5930	. . . 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))
13		imadmrn 5942	. . . 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)⟩)
14	12, 13	eqtr3i 2849	. . 3 ⊢ ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
15	8, 14	syl6eqr 2877	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)))
16		funmpt 6396	. . 3 ⊢ Fun (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
17		dmresexg 5880	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V)
18	17	adantl 484	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V)
19		funimaexg 6443	. . 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)
20	16, 18, 19	sylancr 589	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)) ∈ V)
21	15, 20	eqeltrd 2916	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3497  ⟨cop 4576   ↦ cmpt 5149  dom cdm 5558  ran crn 5559   ↾ cres 5560   “ cima 5561  Fun wfun 6352   Fn wfn 6353  ‘cfv 6358

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366

This theorem is referenced by:  resiexd  6982  fnex  6983  ofexg  7416  cofunexg  7653  dfac8alem  9458  dfac12lem1  9572  cfsmolem  9695  alephsing  9701  itunifval  9841  zorn2lem1  9921  ttukeylem3  9936  imadomg  9959  wunex2  10163  inar1  10200  axdc4uzlem  13354  hashf1rn  13716  bpolylem  15405  1stf1  17445  1stf2  17446  2ndf1  17448  2ndf2  17449  1stfcl  17450  2ndfcl  17451  gsumzadd  19045  satf  32604  frrlem13  33139  madeval  33293  tendo02  37927  dnnumch1  39650  aomclem6  39665  dfrngc2  44250  dfringc2  44296  rngcresringcat  44308  fdivval  44606