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Theorem fex2 7867
Description: A function with bounded domain and codomain is a set. This version of fex 7173 is proven without the Axiom of Replacement ax-rep 5241, but depends on ax-un 7669, which is not required for the proof of fex 7173. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
fex2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)

Proof of Theorem fex2
StepHypRef Expression
1 xpexg 7681 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
213adant1 1130 . 2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
3 fssxp 6694 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
433ad2ant1 1133 . 2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ⊆ (𝐴 × 𝐵))
52, 4ssexd 5280 1 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2106  Vcvv 3444  wss 3909   × cxp 5630  wf 6490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-fun 6496  df-fn 6497  df-f 6498
This theorem is referenced by:  elmapg  8775  f1oen2g  8905  f1dom2g  8906  f1dom2gOLD  8907  dom3d  8931  domssex2  9078  domssex  9079  mapxpen  9084  oismo  9473  wdomima2g  9519  dfac8clem  9965  acni2  9979  acnlem  9981  dfac4  10055  dfac2a  10062  axdc2lem  10381  axdc4lem  10388  axcclem  10390  addex  12910  mulex  12911  seqf1olem2  13945  seqf1o  13946  limsuple  15357  limsuplt  15358  limsupbnd1  15361  caucvgrlem  15554  prdsplusg  17337  prdsmulr  17338  prdsvsca  17339  prdshom  17346  gsumval  18529  frmdplusg  18661  odinf  19341  staffval  20302  cnfldcj  20799  cnfldds  20802  xrsadd  20810  xrsmul  20811  xrsds  20836  ocvfval  21066  cnpfval  22581  iscnp2  22586  fmf  23292  tsmsval  23478  blfvalps  23732  nmfval  23940  tngnm  24011  tngngp2  24012  tngngpd  24013  tngngp  24014  nmoffn  24071  nmofval  24074  ishtpy  24331  tcphex  24577  adjeu  30729  ismeas  32689  isismty  36249  rrnval  36275
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