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Theorem resfunexg 6198
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 5672 . . . . . . 7 (Fun 𝐴 → Fun (𝐴𝐵))
21adantr 474 . . . . . 6 ((Fun 𝐴𝐵𝐶) → Fun (𝐴𝐵))
3 funfn 5662 . . . . . 6 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
42, 3sylib 203 . . . . 5 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) Fn dom (𝐴𝐵))
5 dffn5 5975 . . . . 5 ((𝐴𝐵) Fn dom (𝐴𝐵) ↔ (𝐴𝐵) = (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)))
64, 5sylib 203 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)))
7 fvex 5937 . . . . 5 ((𝐴𝐵)‘𝑥) ∈ V
87fnasrn 6138 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↦ ((𝐴𝐵)‘𝑥)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
96, 8syl6eq 2555 . . 3 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩))
10 opex 4705 . . . . . 6 𝑥, ((𝐴𝐵)‘𝑥)⟩ ∈ V
11 eqid 2505 . . . . . 6 (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) = (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
1210, 11dmmpti 5762 . . . . 5 dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) = dom (𝐴𝐵)
1312imaeq2i 5216 . . . 4 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵))
14 imadmrn 5228 . . . 4 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
1513, 14eqtr3i 2529 . . 3 ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) = ran (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
169, 15syl6eqr 2557 . 2 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) = ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)))
17 funmpt 5669 . . 3 Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩)
18 dmresexg 5177 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
1918adantl 475 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
20 funimaexg 5715 . . 3 ((Fun (𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) ∧ dom (𝐴𝐵) ∈ V) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2117, 19, 20sylancr 685 . 2 ((Fun 𝐴𝐵𝐶) → ((𝑥 ∈ dom (𝐴𝐵) ↦ ⟨𝑥, ((𝐴𝐵)‘𝑥)⟩) “ dom (𝐴𝐵)) ∈ V)
2216, 21eqeltrd 2583 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 378   = wceq 1468  wcel 1937  Vcvv 3066  cop 4001  cmpt 4493  dom cdm 4880  ran crn 4881  cres 4882  cima 4883  Fun wfun 5627   Fn wfn 5628  cfv 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-rep 4548  ax-sep 4558  ax-nul 4567  ax-pr 4680
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3758  df-if 3909  df-sn 3996  df-pr 3998  df-op 4002  df-uni 4229  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-id 4795  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641
This theorem is referenced by:  resiexd  6199  fnex  6200  ofexg  6611  cofunexg  6834  dfac8alem  8545  dfac12lem1  8658  cfsmolem  8785  alephsing  8791  itunifval  8931  zorn2lem1  9011  ttukeylem3  9026  imadomg  9047  wunex2  9248  inar1  9285  axdc4uzlem  12309  hashf1rn  12651  hashf1rnOLD  12652  bpolylem  14261  1stf1  16243  1stf2  16244  2ndf1  16246  2ndf2  16247  1stfcl  16248  2ndfcl  16249  gsumzadd  17716  tendo02  34594  dnnumch1  36142  aomclem6  36157  1wlkreslem  40278  dfrngc2  41164  dfringc2  41210  rngcresringcat  41222  fdivval  41539
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