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Theorem mptfvex 5288
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 2912 . . 3  |-  ( y  =  C  ->  [_ y  /  x ]_ B  = 
[_ C  /  x ]_ B )
2 fvmpt2.1 . . . 4  |-  F  =  ( x  e.  A  |->  B )
3 nfcv 2220 . . . . 5  |-  F/_ y B
4 nfcsb1v 2939 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 2917 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 3880 . . . 4  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
72, 6eqtri 2102 . . 3  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
81, 7fvmptss2 5279 . 2  |-  ( F `
 C )  C_  [_ C  /  x ]_ B
9 elex 2611 . . . . . 6  |-  ( B  e.  V  ->  B  e.  _V )
109alimi 1385 . . . . 5  |-  ( A. x  B  e.  V  ->  A. x  B  e. 
_V )
113nfel1 2230 . . . . . 6  |-  F/ y  B  e.  _V
124nfel1 2230 . . . . . 6  |-  F/ x [_ y  /  x ]_ B  e.  _V
135eleq1d 2148 . . . . . 6  |-  ( x  =  y  ->  ( B  e.  _V  <->  [_ y  /  x ]_ B  e.  _V ) )
1411, 12, 13cbval 1678 . . . . 5  |-  ( A. x  B  e.  _V  <->  A. y [_ y  /  x ]_ B  e.  _V )
1510, 14sylib 120 . . . 4  |-  ( A. x  B  e.  V  ->  A. y [_ y  /  x ]_ B  e. 
_V )
161eleq1d 2148 . . . . 5  |-  ( y  =  C  ->  ( [_ y  /  x ]_ B  e.  _V  <->  [_ C  /  x ]_ B  e.  _V )
)
1716spcgv 2686 . . . 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 123 . 2  |-  ( ( A. x  B  e.  V  /\  C  e.  W )  ->  [_ C  /  x ]_ B  e. 
_V )
20 ssexg 3925 . 2  |-  ( ( ( F `  C
)  C_  [_ C  /  x ]_ B  /\  [_ C  /  x ]_ B  e. 
_V )  ->  ( F `  C )  e.  _V )
218, 19, 20sylancr 405 1  |-  ( ( A. x  B  e.  V  /\  C  e.  W )  ->  ( F `  C )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283    = wceq 1285    e. wcel 1434   _Vcvv 2602   [_csb 2909    C_ wss 2974    |-> cmpt 3847   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-iota 4897  df-fun 4934  df-fv 4940
This theorem is referenced by:  mpt2fvex  5860  xpcomco  6370
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