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Theorem mptexgf 27689
Description: If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) (Revised by Thierry Arnoux, 17-May-2020.)
Hypothesis
Ref Expression
mptexgf.a  |-  F/_ x A
Assertion
Ref Expression
mptexgf  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )

Proof of Theorem mptexgf
StepHypRef Expression
1 funmpt 5606 . 2  |-  Fun  (
x  e.  A  |->  B )
2 eqid 2454 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32dmmpt 5485 . . . 4  |-  dom  (
x  e.  A  |->  B )  =  { x  e.  A  |  B  e.  _V }
4 a1tru 1414 . . . . . . 7  |-  ( B  e.  _V  -> T.  )
54rgenw 2815 . . . . . 6  |-  A. x  e.  A  ( B  e.  _V  -> T.  )
6 ss2rab 3562 . . . . . 6  |-  ( { x  e.  A  |  B  e.  _V }  C_  { x  e.  A  | T.  }  <->  A. x  e.  A  ( B  e.  _V  -> T.  ) )
75, 6mpbir 209 . . . . 5  |-  { x  e.  A  |  B  e.  _V }  C_  { x  e.  A  | T.  }
8 mptexgf.a . . . . . 6  |-  F/_ x A
98rabtru 27601 . . . . 5  |-  { x  e.  A  | T.  }  =  A
107, 9sseqtri 3521 . . . 4  |-  { x  e.  A  |  B  e.  _V }  C_  A
113, 10eqsstri 3519 . . 3  |-  dom  (
x  e.  A  |->  B )  C_  A
12 ssexg 4583 . . 3  |-  ( ( dom  ( x  e.  A  |->  B )  C_  A  /\  A  e.  V
)  ->  dom  ( x  e.  A  |->  B )  e.  _V )
1311, 12mpan 668 . 2  |-  ( A  e.  V  ->  dom  ( x  e.  A  |->  B )  e.  _V )
14 funex 6115 . 2  |-  ( ( Fun  ( x  e.  A  |->  B )  /\  dom  ( x  e.  A  |->  B )  e.  _V )  ->  ( x  e.  A  |->  B )  e. 
_V )
151, 13, 14sylancr 661 1  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   T. wtru 1399    e. wcel 1823   F/_wnfc 2602   A.wral 2804   {crab 2808   _Vcvv 3106    C_ wss 3461    |-> cmpt 4497   dom cdm 4988   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578
This theorem is referenced by:  esumrnmpt2  28300
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