Theorem resfunexg 5738

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 5258	. . . . 5 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵))
2		funfvex 5533	. . . . . 6 ⊢ ((Fun (𝐴 ↾ 𝐵) ∧ 𝑥 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑥) ∈ V)
3	2	ralrimiva 2550	. . . . 5 ⊢ (Fun (𝐴 ↾ 𝐵) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)((𝐴 ↾ 𝐵)‘𝑥) ∈ V)
4		fnasrng 5697	. . . . 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 5247	. . . . 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 5562	. . . 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 2741	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13		opexg 4229	. . . . . . . . 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 2550	. . . . . . 7 ⊢ (Fun (𝐴 ↾ 𝐵) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩ ∈ V)
16		dmmptg 5127	. . . . . . 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 4971	. . . . 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 4981	. . . . 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 2225	. . . 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 2220	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)))
23		funmpt 5255	. . 3 ⊢ Fun (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
24		dmresexg 4931	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V)
25	24	adantl 277	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V)
26		funimaexg 5301	. . 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 2254	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2738  ⟨cop 3596   ↦ cmpt 4065  dom cdm 4627  ran crn 4628   ↾ cres 4629   “ cima 4630  Fun wfun 5211   Fn wfn 5212  ‘cfv 5217

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225

This theorem is referenced by:  fnex  5739  ofexg  6087  cofunexg  6110  rdgivallem  6382  frecex  6395  frecsuclem  6407  djudoml  7218  djudomr  7219  fihashf1rn  10768  qnnen  12432