Theorem resfunexg 5303

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 4863	. . . . 5 ⊢ (Fun A → Fun (A ↾ B))
2		funfvex 5113	. . . . . 6 ⊢ ((Fun (A ↾ B) ∧ x ∈ dom (A ↾ B)) → ((A ↾ B)‘x) ∈ V)
3	2	ralrimiva 2366	. . . . 5 ⊢ (Fun (A ↾ B) → ∀x ∈ dom (A ↾ B)((A ↾ B)‘x) ∈ V)
4		fnasrng 5264	. . . . 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 261	. . 3 ⊢ ((Fun A ∧ B ∈ 𝐶) → (x ∈ dom (A ↾ B) ↦ ((A ↾ B)‘x)) = ran (x ∈ dom (A ↾ B) ↦ ⟨x, ((A ↾ B)‘x)⟩))
7	1	adantr 261	. . . . 5 ⊢ ((Fun A ∧ B ∈ 𝐶) → Fun (A ↾ B))
8		funfn 4853	. . . . 5 ⊢ (Fun (A ↾ B) ↔ (A ↾ B) Fn dom (A ↾ B))
9	7, 8	sylib 127	. . . 4 ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ B) Fn dom (A ↾ B))
10		dffn5im 5140	. . . 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		imadmrn 4601	. . . . 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)⟩)
13		vex 2534	. . . . . . . . 9 ⊢ x ∈ V
14		opexgOLD 3935	. . . . . . . . 9 ⊢ ((x ∈ V ∧ ((A ↾ B)‘x) ∈ V) → ⟨x, ((A ↾ B)‘x)⟩ ∈ V)
15	13, 2, 14	sylancr 395	. . . . . . . 8 ⊢ ((Fun (A ↾ B) ∧ x ∈ dom (A ↾ B)) → ⟨x, ((A ↾ B)‘x)⟩ ∈ V)
16	15	ralrimiva 2366	. . . . . . 7 ⊢ (Fun (A ↾ B) → ∀x ∈ dom (A ↾ B)⟨x, ((A ↾ B)‘x)⟩ ∈ V)
17		dmmptg 4741	. . . . . . 7 ⊢ (∀x ∈ dom (A ↾ B)⟨x, ((A ↾ B)‘x)⟩ ∈ V → dom (x ∈ dom (A ↾ B) ↦ ⟨x, ((A ↾ B)‘x)⟩) = dom (A ↾ B))
18	1, 16, 17	3syl 17	. . . . . 6 ⊢ (Fun A → dom (x ∈ dom (A ↾ B) ↦ ⟨x, ((A ↾ B)‘x)⟩) = dom (A ↾ B))
19	18	imaeq2d 4591	. . . . 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)))
20	12, 19	syl5reqr 2065	. . . 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 261	. . 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 2060	. 2 ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ B) = ((x ∈ dom (A ↾ B) ↦ ⟨x, ((A ↾ B)‘x)⟩) “ dom (A ↾ B)))
23		funmpt 4860	. . 3 ⊢ Fun (x ∈ dom (A ↾ B) ↦ ⟨x, ((A ↾ B)‘x)⟩)
24		dmresexg 4557	. . . 4 ⊢ (B ∈ 𝐶 → dom (A ↾ B) ∈ V)
25	24	adantl 262	. . 3 ⊢ ((Fun A ∧ B ∈ 𝐶) → dom (A ↾ B) ∈ V)
26		funimaexg 4905	. . 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 395	. 2 ⊢ ((Fun A ∧ B ∈ 𝐶) → ((x ∈ dom (A ↾ B) ↦ ⟨x, ((A ↾ B)‘x)⟩) “ dom (A ↾ B)) ∈ V)
28	22, 27	eqeltrd 2092	1 ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  ∀wral 2280  Vcvv 2531  ⟨cop 3349   ↦ cmpt 3788  dom cdm 4268  ran crn 4269   ↾ cres 4270   “ cima 4271  Fun wfun 4819   Fn wfn 4820  ‘cfv 4825

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833

This theorem is referenced by:  fnex  5304  ofexg  5635  cofunexg  5657  rdgivallem  5884  frecsuclem3  5902