Theorem resfunexg 5325

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 4884	. . . . 5 ⊢ (Fun A → Fun (A ↾ B))
2		funfvex 5135	. . . . . 6 ⊢ ((Fun (A ↾ B) ∧ x ∈ dom (A ↾ B)) → ((A ↾ B)‘x) ∈ V)
3	2	ralrimiva 2386	. . . . 5 ⊢ (Fun (A ↾ B) → ∀x ∈ dom (A ↾ B)((A ↾ B)‘x) ∈ V)
4		fnasrng 5286	. . . . 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 4874	. . . . 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 5162	. . . 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 4621	. . . . 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 2554	. . . . . . . . 9 ⊢ x ∈ V
14		opexgOLD 3956	. . . . . . . . 9 ⊢ ((x ∈ V ∧ ((A ↾ B)‘x) ∈ V) → ⟨x, ((A ↾ B)‘x)⟩ ∈ V)
15	13, 2, 14	sylancr 393	. . . . . . . 8 ⊢ ((Fun (A ↾ B) ∧ x ∈ dom (A ↾ B)) → ⟨x, ((A ↾ B)‘x)⟩ ∈ V)
16	15	ralrimiva 2386	. . . . . . 7 ⊢ (Fun (A ↾ B) → ∀x ∈ dom (A ↾ B)⟨x, ((A ↾ B)‘x)⟩ ∈ V)
17		dmmptg 4761	. . . . . . 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 4611	. . . . 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 2084	. . . 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 2079	. 2 ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ B) = ((x ∈ dom (A ↾ B) ↦ ⟨x, ((A ↾ B)‘x)⟩) “ dom (A ↾ B)))
23		funmpt 4881	. . 3 ⊢ Fun (x ∈ dom (A ↾ B) ↦ ⟨x, ((A ↾ B)‘x)⟩)
24		dmresexg 4577	. . . 4 ⊢ (B ∈ 𝐶 → dom (A ↾ B) ∈ V)
25	24	adantl 262	. . 3 ⊢ ((Fun A ∧ B ∈ 𝐶) → dom (A ↾ B) ∈ V)
26		funimaexg 4926	. . 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 393	. 2 ⊢ ((Fun A ∧ B ∈ 𝐶) → ((x ∈ dom (A ↾ B) ↦ ⟨x, ((A ↾ B)‘x)⟩) “ dom (A ↾ B)) ∈ V)
28	22, 27	eqeltrd 2111	1 ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551  ⟨cop 3370   ↦ cmpt 3809  dom cdm 4288  ran crn 4289   ↾ cres 4290   “ cima 4291  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853

This theorem is referenced by:  fnex  5326  ofexg  5658  cofunexg  5680  rdgivallem  5908  frecsuclem3  5929