ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resfunexg GIF version

Theorem resfunexg 5434
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 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Proof of Theorem resfunexg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funres 4991 . . . . 5 (Fun 𝐴 → Fun (𝐴𝐵))
2 funfvex 5243 . . . . . 6 ((Fun (𝐴𝐵) ∧ 𝑥 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑥) ∈ V)
32ralrimiva 2439 . . . . 5 (Fun (𝐴𝐵) → ∀𝑥 ∈ dom (𝐴𝐵)((𝐴𝐵)‘𝑥) ∈ V)
4 fnasrng 5395 . . . . 5 (∀𝑥 ∈ dom (𝐴𝐵)((𝐴𝐵)‘𝑥) ∈ V → (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
51, 3, 43syl 17 . . . 4 (Fun 𝐴 → (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
65adantr 270 . . 3 ((Fun 𝐴𝐵𝐶) → (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
71adantr 270 . . . . 5 ((Fun 𝐴𝐵𝐶) → Fun (𝐴𝐵))
8 funfn 4981 . . . . 5 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
97, 8sylib 120 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) Fn dom (𝐴𝐵))
10 dffn5im 5271 . . . 4 ((𝐴𝐵) Fn dom (𝐴𝐵) → (𝐴𝐵) = (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)))
119, 10syl 14 . . 3 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)))
12 imadmrn 4728 . . . . 5 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
13 vex 2613 . . . . . . . . 9 𝑥 ∈ V
14 opexg 4011 . . . . . . . . 9 ((𝑥 ∈ V ∧ ((𝐴𝐵)‘𝑥) ∈ V) → ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
1513, 2, 14sylancr 405 . . . . . . . 8 ((Fun (𝐴𝐵) ∧ 𝑥 ∈ dom (𝐴𝐵)) → ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
1615ralrimiva 2439 . . . . . . 7 (Fun (𝐴𝐵) → ∀𝑥 ∈ dom (𝐴𝐵)⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
17 dmmptg 4868 . . . . . . 7 (∀𝑥 ∈ dom (𝐴𝐵)⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V → dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) = dom (𝐴𝐵))
181, 16, 173syl 17 . . . . . 6 (Fun 𝐴 → dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) = dom (𝐴𝐵))
1918imaeq2d 4718 . . . . 5 (Fun 𝐴 → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)))
2012, 19syl5reqr 2130 . . . 4 (Fun 𝐴 → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
2120adantr 270 . . 3 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
226, 11, 213eqtr4d 2125 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)))
23 funmpt 4988 . . 3 Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
24 dmresexg 4682 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
2524adantl 271 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
26 funimaexg 5034 . . 3 ((Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) ∧ dom (𝐴𝐵) ∈ V) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2723, 25, 26sylancr 405 . 2 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2822, 27eqeltrd 2159 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  wral 2353  Vcvv 2610  cop 3419  cmpt 3859  dom cdm 4391  ran crn 4392  cres 4393  cima 4394  Fun wfun 4946   Fn wfn 4947  cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960
This theorem is referenced by:  fnex  5435  ofexg  5767  cofunexg  5789  rdgivallem  6050  frecex  6063  frecsuclem  6075  fihashf1rn  9865
  Copyright terms: Public domain W3C validator