Theorem resfunexg 5761

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 5279	. . . . 5 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵))
2		funfvex 5554	. . . . . 6 ⊢ ((Fun (𝐴 ↾ 𝐵) ∧ 𝑥 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑥) ∈ V)
3	2	ralrimiva 2563	. . . . 5 ⊢ (Fun (𝐴 ↾ 𝐵) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)((𝐴 ↾ 𝐵)‘𝑥) ∈ V)
4		fnasrng 5720	. . . . 5 ⊢ (∀𝑥 ∈ dom (𝐴 ↾ 𝐵)((𝐴 ↾ 𝐵)‘𝑥) ∈ V → (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩))
5	1, 3, 4	3syl 17	. . . 4 ⊢ (Fun 𝐴 → (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩))
6	5	adantr 276	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩))
7	1	adantr 276	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → Fun (𝐴 ↾ 𝐵))
8		funfn 5268	. . . . 5 ⊢ (Fun (𝐴 ↾ 𝐵) ↔ (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
9	7, 8	sylib 122	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵))
10		dffn5im 5585	. . . 4 ⊢ ((𝐴 ↾ 𝐵) Fn dom (𝐴 ↾ 𝐵) → (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)))
11	9, 10	syl 14	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ((𝐴 ↾ 𝐵)‘𝑥)))
12		vex 2755	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13		opexg 4249	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ ((𝐴 ↾ 𝐵)‘𝑥) ∈ V) → ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩ ∈ V)
14	12, 2, 13	sylancr 414	. . . . . . . 8 ⊢ ((Fun (𝐴 ↾ 𝐵) ∧ 𝑥 ∈ dom (𝐴 ↾ 𝐵)) → ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩ ∈ V)
15	14	ralrimiva 2563	. . . . . . 7 ⊢ (Fun (𝐴 ↾ 𝐵) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩ ∈ V)
16		dmmptg 5147	. . . . . . 7 ⊢ (∀𝑥 ∈ dom (𝐴 ↾ 𝐵)⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩ ∈ V → dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) = dom (𝐴 ↾ 𝐵))
17	1, 15, 16	3syl 17	. . . . . 6 ⊢ (Fun 𝐴 → dom (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) = dom (𝐴 ↾ 𝐵))
18	17	imaeq2d 4991	. . . . 5 ⊢ (Fun 𝐴 → ((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)))
19		imadmrn 5001	. . . . 5 ⊢ ((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)⟩)
20	18, 19	eqtr3di 2237	. . . 4 ⊢ (Fun 𝐴 → ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩))
21	20	adantr 276	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)) = ran (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩))
22	6, 11, 21	3eqtr4d 2232	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)))
23		funmpt 5276	. . 3 ⊢ Fun (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
24		dmresexg 4951	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V)
25	24	adantl 277	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V)
26		funimaexg 5322	. . 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)
27	23, 25, 26	sylancr 414	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)) ∈ V)
28	22, 27	eqeltrd 2266	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  ∀wral 2468  Vcvv 2752  ⟨cop 3613   ↦ cmpt 4082  dom cdm 4647  ran crn 4648   ↾ cres 4649   “ cima 4650  Fun wfun 5232   Fn wfn 5233  ‘cfv 5238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246

This theorem is referenced by:  fnex  5762  ofexg  6115  cofunexg  6138  rdgivallem  6410  frecex  6423  frecsuclem  6435  djudoml  7253  djudomr  7254  fihashf1rn  10809  qnnen  12493