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Theorem fex2 5396
Description: A function with bounded domain and codomain is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
fex2  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )

Proof of Theorem fex2
StepHypRef Expression
1 xpexg 4752 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
213adant1 1016 . 2  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
3 fssxp 5395 . . 3  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
433ad2ant1 1019 . 2  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  C_  ( A  X.  B ) )
52, 4ssexd 4155 1  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 979    e. wcel 2158   _Vcvv 2749    C_ wss 3141    X. cxp 4636   -->wf 5224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-fun 5230  df-fn 5231  df-f 5232
This theorem is referenced by:  elmapg  6675  f1oen2g  6769  f1dom2g  6770  dom3d  6788  mapxpen  6862  addex  9665  mulex  9666  climrecvg1n  11370  cnpfval  14048  txcn  14128  blfvalps  14238
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