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Theorem resfunexg 5609
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 5134 . . . . 5 (Fun 𝐴 → Fun (𝐴𝐵))
2 funfvex 5406 . . . . . 6 ((Fun (𝐴𝐵) ∧ 𝑥 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑥) ∈ V)
32ralrimiva 2482 . . . . 5 (Fun (𝐴𝐵) → ∀𝑥 ∈ dom (𝐴𝐵)((𝐴𝐵)‘𝑥) ∈ V)
4 fnasrng 5568 . . . . 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 274 . . 3 ((Fun 𝐴𝐵𝐶) → (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
71adantr 274 . . . . 5 ((Fun 𝐴𝐵𝐶) → Fun (𝐴𝐵))
8 funfn 5123 . . . . 5 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
97, 8sylib 121 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) Fn dom (𝐴𝐵))
10 dffn5im 5435 . . . 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 4861 . . . . 5 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
13 vex 2663 . . . . . . . . 9 𝑥 ∈ V
14 opexg 4120 . . . . . . . . 9 ((𝑥 ∈ V ∧ ((𝐴𝐵)‘𝑥) ∈ V) → ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
1513, 2, 14sylancr 410 . . . . . . . 8 ((Fun (𝐴𝐵) ∧ 𝑥 ∈ dom (𝐴𝐵)) → ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
1615ralrimiva 2482 . . . . . . 7 (Fun (𝐴𝐵) → ∀𝑥 ∈ dom (𝐴𝐵)⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
17 dmmptg 5006 . . . . . . 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 4851 . . . . 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 2165 . . . 4 (Fun 𝐴 → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
2120adantr 274 . . 3 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
226, 11, 213eqtr4d 2160 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)))
23 funmpt 5131 . . 3 Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
24 dmresexg 4812 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
2524adantl 275 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
26 funimaexg 5177 . . 3 ((Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) ∧ dom (𝐴𝐵) ∈ V) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2723, 25, 26sylancr 410 . 2 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2822, 27eqeltrd 2194 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  wral 2393  Vcvv 2660  cop 3500  cmpt 3959  dom cdm 4509  ran crn 4510  cres 4511  cima 4512  Fun wfun 5087   Fn wfn 5088  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by:  fnex  5610  ofexg  5954  cofunexg  5977  rdgivallem  6246  frecex  6259  frecsuclem  6271  djudoml  7043  djudomr  7044  fihashf1rn  10503  qnnen  11871
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