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Theorem fex2 5192
Description: A function with bounded domain and range 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 4565 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
213adant1 962 . 2  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
3 fssxp 5191 . . 3  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
433ad2ant1 965 . 2  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  C_  ( A  X.  B ) )
52, 4ssexd 3985 1  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 925    e. wcel 1439   _Vcvv 2620    C_ wss 3000    X. cxp 4449   -->wf 5024
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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-xp 4457  df-rel 4458  df-cnv 4459  df-dm 4461  df-rn 4462  df-fun 5030  df-fn 5031  df-f 5032
This theorem is referenced by:  elmapg  6432  f1oen2g  6526  f1dom2g  6527  dom3d  6545  mapxpen  6618  climrecvg1n  10791
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