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Theorem resfunexg 5761
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 5279 . . . . 5 (Fun 𝐴 → Fun (𝐴𝐵))
2 funfvex 5554 . . . . . 6 ((Fun (𝐴𝐵) ∧ 𝑥 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑥) ∈ V)
32ralrimiva 2563 . . . . 5 (Fun (𝐴𝐵) → ∀𝑥 ∈ dom (𝐴𝐵)((𝐴𝐵)‘𝑥) ∈ V)
4 fnasrng 5720 . . . . 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 276 . . 3 ((Fun 𝐴𝐵𝐶) → (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
71adantr 276 . . . . 5 ((Fun 𝐴𝐵𝐶) → Fun (𝐴𝐵))
8 funfn 5268 . . . . 5 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
97, 8sylib 122 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) Fn dom (𝐴𝐵))
10 dffn5im 5585 . . . 4 ((𝐴𝐵) Fn dom (𝐴𝐵) → (𝐴𝐵) = (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)))
119, 10syl 14 . . 3 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)))
12 vex 2755 . . . . . . . . 9 𝑥 ∈ V
13 opexg 4249 . . . . . . . . 9 ((𝑥 ∈ V ∧ ((𝐴𝐵)‘𝑥) ∈ V) → ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
1412, 2, 13sylancr 414 . . . . . . . 8 ((Fun (𝐴𝐵) ∧ 𝑥 ∈ dom (𝐴𝐵)) → ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
1514ralrimiva 2563 . . . . . . 7 (Fun (𝐴𝐵) → ∀𝑥 ∈ dom (𝐴𝐵)⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
16 dmmptg 5147 . . . . . . 7 (∀𝑥 ∈ dom (𝐴𝐵)⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V → dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) = dom (𝐴𝐵))
171, 15, 163syl 17 . . . . . 6 (Fun 𝐴 → dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) = dom (𝐴𝐵))
1817imaeq2d 4991 . . . . 5 (Fun 𝐴 → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)))
19 imadmrn 5001 . . . . 5 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
2018, 19eqtr3di 2237 . . . 4 (Fun 𝐴 → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
2120adantr 276 . . 3 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
226, 11, 213eqtr4d 2232 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)))
23 funmpt 5276 . . 3 Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
24 dmresexg 4951 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
2524adantl 277 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
26 funimaexg 5322 . . 3 ((Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) ∧ dom (𝐴𝐵) ∈ V) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2723, 25, 26sylancr 414 . 2 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2822, 27eqeltrd 2266 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468  Vcvv 2752  cop 3613  cmpt 4082  dom cdm 4647  ran crn 4648  cres 4649  cima 4650  Fun wfun 5232   Fn wfn 5233  cfv 5238
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246
This theorem is referenced by:  fnex  5762  ofexg  6115  cofunexg  6138  rdgivallem  6410  frecex  6423  frecsuclem  6435  djudoml  7253  djudomr  7254  fihashf1rn  10809  qnnen  12493
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