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Theorem mptfvex 5617
Description: Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypothesis
Ref Expression
fvmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
mptfvex  |-  ( ( A. x  B  e.  V  /\  C  e.  W )  ->  ( F `  C )  e.  _V )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    B( x)    F( x)    V( x)    W( x)

Proof of Theorem mptfvex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3075 . . 3  |-  ( y  =  C  ->  [_ y  /  x ]_ B  = 
[_ C  /  x ]_ B )
2 fvmpt2.1 . . . 4  |-  F  =  ( x  e.  A  |->  B )
3 nfcv 2332 . . . . 5  |-  F/_ y B
4 nfcsb1v 3105 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 3081 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 4113 . . . 4  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
72, 6eqtri 2210 . . 3  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
81, 7fvmptss2 5607 . 2  |-  ( F `
 C )  C_  [_ C  /  x ]_ B
9 elex 2763 . . . . . 6  |-  ( B  e.  V  ->  B  e.  _V )
109alimi 1466 . . . . 5  |-  ( A. x  B  e.  V  ->  A. x  B  e. 
_V )
113nfel1 2343 . . . . . 6  |-  F/ y  B  e.  _V
124nfel1 2343 . . . . . 6  |-  F/ x [_ y  /  x ]_ B  e.  _V
135eleq1d 2258 . . . . . 6  |-  ( x  =  y  ->  ( B  e.  _V  <->  [_ y  /  x ]_ B  e.  _V ) )
1411, 12, 13cbval 1765 . . . . 5  |-  ( A. x  B  e.  _V  <->  A. y [_ y  /  x ]_ B  e.  _V )
1510, 14sylib 122 . . . 4  |-  ( A. x  B  e.  V  ->  A. y [_ y  /  x ]_ B  e. 
_V )
161eleq1d 2258 . . . . 5  |-  ( y  =  C  ->  ( [_ y  /  x ]_ B  e.  _V  <->  [_ C  /  x ]_ B  e.  _V )
)
1716spcgv 2839 . . . 4  |-  ( C  e.  W  ->  ( A. y [_ y  /  x ]_ B  e.  _V  ->  [_ C  /  x ]_ B  e.  _V ) )
1815, 17syl5 32 . . 3  |-  ( C  e.  W  ->  ( A. x  B  e.  V  ->  [_ C  /  x ]_ B  e.  _V ) )
1918impcom 125 . 2  |-  ( ( A. x  B  e.  V  /\  C  e.  W )  ->  [_ C  /  x ]_ B  e. 
_V )
20 ssexg 4157 . 2  |-  ( ( ( F `  C
)  C_  [_ C  /  x ]_ B  /\  [_ C  /  x ]_ B  e. 
_V )  ->  ( F `  C )  e.  _V )
218, 19, 20sylancr 414 1  |-  ( ( A. x  B  e.  V  /\  C  e.  W )  ->  ( F `  C )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1362    = wceq 1364    e. wcel 2160   _Vcvv 2752   [_csb 3072    C_ wss 3144    |-> cmpt 4079   ` cfv 5231
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-iota 5193  df-fun 5233  df-fv 5239
This theorem is referenced by:  mpofvex  6222  xpcomco  6844  lssex  13631
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