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Theorem fnmpt 5174
Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
Hypothesis
Ref Expression
mptfng.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fnmpt (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fnmpt
StepHypRef Expression
1 elex 2644 . . 3 (𝐵𝑉𝐵 ∈ V)
21ralimi 2449 . 2 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
3 mptfng.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptfng 5173 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
52, 4sylib 121 1 (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1296  wcel 1445  wral 2370  Vcvv 2633  cmpt 3921   Fn wfn 5044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-fun 5051  df-fn 5052
This theorem is referenced by:  mpt0  5175  ralrnmpt  5480  rexrnmpt  5481  fmpt  5488  fmpt2d  5499  f1ocnvd  5884  offval2  5908  ofrfval2  5909  caofinvl  5915  f1od2  6038  frectfr  6203  omfnex  6250  oeiv  6257  mptelixpg  6531  efcvgfsum  11121  neif  12008  tgrest  12036  nninfalllemn  12607
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