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Theorem resfunexg 5382
 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 4941 . . . . 5 (Fun 𝐴 → Fun (𝐴𝐵))
2 funfvex 5192 . . . . . 6 ((Fun (𝐴𝐵) ∧ 𝑥 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑥) ∈ V)
32ralrimiva 2392 . . . . 5 (Fun (𝐴𝐵) → ∀𝑥 ∈ dom (𝐴𝐵)((𝐴𝐵)‘𝑥) ∈ V)
4 fnasrng 5343 . . . . 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 261 . . 3 ((Fun 𝐴𝐵𝐶) → (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
71adantr 261 . . . . 5 ((Fun 𝐴𝐵𝐶) → Fun (𝐴𝐵))
8 funfn 4931 . . . . 5 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
97, 8sylib 127 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) Fn dom (𝐴𝐵))
10 dffn5im 5219 . . . 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 4678 . . . . 5 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
13 vex 2560 . . . . . . . . 9 𝑥 ∈ V
14 opexgOLD 3965 . . . . . . . . 9 ((𝑥 ∈ V ∧ ((𝐴𝐵)‘𝑥) ∈ V) → ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
1513, 2, 14sylancr 393 . . . . . . . 8 ((Fun (𝐴𝐵) ∧ 𝑥 ∈ dom (𝐴𝐵)) → ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
1615ralrimiva 2392 . . . . . . 7 (Fun (𝐴𝐵) → ∀𝑥 ∈ dom (𝐴𝐵)⟨𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V)
17 dmmptg 4818 . . . . . . 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 4668 . . . . 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 2087 . . . 4 (Fun 𝐴 → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
2120adantr 261 . . 3 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
226, 11, 213eqtr4d 2082 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)))
23 funmpt 4938 . . 3 Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
24 dmresexg 4634 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
2524adantl 262 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
26 funimaexg 4983 . . 3 ((Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) ∧ dom (𝐴𝐵) ∈ V) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2723, 25, 26sylancr 393 . 2 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2822, 27eqeltrd 2114 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  Vcvv 2557  ⟨cop 3378   ↦ cmpt 3818  dom cdm 4345  ran crn 4346   ↾ cres 4347   “ cima 4348  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910 This theorem is referenced by:  fnex  5383  ofexg  5716  cofunexg  5738  rdgivallem  5968  frecex  5981  frecsuclem3  5990
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