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Theorem resfunexg 6362
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 5829 . . . . . . 7 (Fun 𝐴 → Fun (𝐴𝐵))
21adantr 479 . . . . . 6 ((Fun 𝐴𝐵𝐶) → Fun (𝐴𝐵))
3 funfn 5819 . . . . . 6 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
42, 3sylib 206 . . . . 5 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) Fn dom (𝐴𝐵))
5 dffn5 6136 . . . . 5 ((𝐴𝐵) Fn dom (𝐴𝐵) ↔ (𝐴𝐵) = (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)))
64, 5sylib 206 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)))
7 fvex 6098 . . . . 5 ((𝐴𝐵)‘𝑥) ∈ V
87fnasrn 6302 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
96, 8syl6eq 2659 . . 3 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
10 opex 4853 . . . . . 6 𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V
11 eqid 2609 . . . . . 6 (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) = (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
1210, 11dmmpti 5922 . . . . 5 dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) = dom (𝐴𝐵)
1312imaeq2i 5370 . . . 4 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵))
14 imadmrn 5382 . . . 4 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
1513, 14eqtr3i 2633 . . 3 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
169, 15syl6eqr 2661 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)))
17 funmpt 5826 . . 3 Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
18 dmresexg 5328 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
1918adantl 480 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
20 funimaexg 5875 . . 3 ((Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) ∧ dom (𝐴𝐵) ∈ V) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2117, 19, 20sylancr 693 . 2 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2216, 21eqeltrd 2687 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cop 4130  cmpt 4637  dom cdm 5028  ran crn 5029  cres 5030  cima 5031  Fun wfun 5784   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798
This theorem is referenced by:  resiexd  6363  fnex  6364  ofexg  6776  cofunexg  7000  dfac8alem  8712  dfac12lem1  8825  cfsmolem  8952  alephsing  8958  itunifval  9098  zorn2lem1  9178  ttukeylem3  9193  imadomg  9214  wunex2  9416  inar1  9453  axdc4uzlem  12599  hashf1rn  12956  hashf1rnOLD  12957  bpolylem  14564  1stf1  16601  1stf2  16602  2ndf1  16604  2ndf2  16605  1stfcl  16606  2ndfcl  16607  gsumzadd  18091  tendo02  34896  dnnumch1  36435  aomclem6  36450  1wlkreslem  40880  dfrngc2  41766  dfringc2  41812  rngcresringcat  41824  fdivval  42133
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