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Theorem fex2 7871
Description: A function with bounded domain and codomain is a set. This version of fex 7177 is proven without the Axiom of Replacement ax-rep 5243, but depends on ax-un 7673, which is not required for the proof of fex 7177. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
fex2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)

Proof of Theorem fex2
StepHypRef Expression
1 xpexg 7685 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
213adant1 1131 . 2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
3 fssxp 6697 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
433ad2ant1 1134 . 2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ⊆ (𝐴 × 𝐵))
52, 4ssexd 5282 1 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088  wcel 2107  Vcvv 3444  wss 3911   × cxp 5632  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  elmapg  8781  f1oen2g  8911  f1dom2g  8912  f1dom2gOLD  8913  dom3d  8937  domssex2  9084  domssex  9085  mapxpen  9090  oismo  9481  wdomima2g  9527  dfac8clem  9973  acni2  9987  acnlem  9989  dfac4  10063  dfac2a  10070  axdc2lem  10389  axdc4lem  10396  axcclem  10398  addex  12918  mulex  12919  seqf1olem2  13954  seqf1o  13955  limsuple  15366  limsuplt  15367  limsupbnd1  15370  caucvgrlem  15563  prdsplusg  17345  prdsmulr  17346  prdsvsca  17347  prdshom  17354  gsumval  18537  frmdplusg  18669  odinf  19350  staffval  20320  cnfldcj  20819  cnfldds  20822  xrsadd  20830  xrsmul  20831  xrsds  20856  ocvfval  21086  cnpfval  22601  iscnp2  22606  fmf  23312  tsmsval  23498  blfvalps  23752  nmfval  23960  tngnm  24031  tngngp2  24032  tngngpd  24033  tngngp  24034  nmoffn  24091  nmofval  24094  ishtpy  24351  tcphex  24597  adjeu  30873  ismeas  32855  isismty  36306  rrnval  36332
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