Theorem resfunexg 5812

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 5317	. . . . 5 ⊢ (Fun 𝐴 → Fun (𝐴 ↾ 𝐵))
2		funfvex 5600	. . . . . 6 ⊢ ((Fun (𝐴 ↾ 𝐵) ∧ 𝑥 ∈ dom (𝐴 ↾ 𝐵)) → ((𝐴 ↾ 𝐵)‘𝑥) ∈ V)
3	2	ralrimiva 2580	. . . . 5 ⊢ (Fun (𝐴 ↾ 𝐵) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)((𝐴 ↾ 𝐵)‘𝑥) ∈ V)
4		fnasrng 5767	. . . . 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 5306	. . . . 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 5631	. . . 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 2776	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13		opexg 4276	. . . . . . . . 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 2580	. . . . . . 7 ⊢ (Fun (𝐴 ↾ 𝐵) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐵)⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩ ∈ V)
16		dmmptg 5185	. . . . . . 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 5027	. . . . 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 5037	. . . . 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 2254	. . . 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 2249	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) = ((𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩) “ dom (𝐴 ↾ 𝐵)))
23		funmpt 5314	. . 3 ⊢ Fun (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↦ ⟨𝑥, ((𝐴 ↾ 𝐵)‘𝑥)⟩)
24		dmresexg 4987	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V)
25	24	adantl 277	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V)
26		funimaexg 5363	. . 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 2283	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773  ⟨cop 3637   ↦ cmpt 4109  dom cdm 4679  ran crn 4680   ↾ cres 4681   “ cima 4682  Fun wfun 5270   Fn wfn 5271  ‘cfv 5276

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284

This theorem is referenced by:  fnex  5813  ofexg  6170  cofunexg  6201  rdgivallem  6474  frecex  6487  frecsuclem  6499  djudoml  7338  djudomr  7339  fihashf1rn  10940  qnnen  12846