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Theorem mptfvex 5424
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 2950 . . 3  |-  ( y  =  C  ->  [_ y  /  x ]_ B  = 
[_ C  /  x ]_ B )
2 fvmpt2.1 . . . 4  |-  F  =  ( x  e.  A  |->  B )
3 nfcv 2235 . . . . 5  |-  F/_ y B
4 nfcsb1v 2977 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 2955 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 3955 . . . 4  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
72, 6eqtri 2115 . . 3  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
81, 7fvmptss2 5414 . 2  |-  ( F `
 C )  C_  [_ C  /  x ]_ B
9 elex 2644 . . . . . 6  |-  ( B  e.  V  ->  B  e.  _V )
109alimi 1396 . . . . 5  |-  ( A. x  B  e.  V  ->  A. x  B  e. 
_V )
113nfel1 2246 . . . . . 6  |-  F/ y  B  e.  _V
124nfel1 2246 . . . . . 6  |-  F/ x [_ y  /  x ]_ B  e.  _V
135eleq1d 2163 . . . . . 6  |-  ( x  =  y  ->  ( B  e.  _V  <->  [_ y  /  x ]_ B  e.  _V ) )
1411, 12, 13cbval 1691 . . . . 5  |-  ( A. x  B  e.  _V  <->  A. y [_ y  /  x ]_ B  e.  _V )
1510, 14sylib 121 . . . 4  |-  ( A. x  B  e.  V  ->  A. y [_ y  /  x ]_ B  e. 
_V )
161eleq1d 2163 . . . . 5  |-  ( y  =  C  ->  ( [_ y  /  x ]_ B  e.  _V  <->  [_ C  /  x ]_ B  e.  _V )
)
1716spcgv 2720 . . . 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 124 . 2  |-  ( ( A. x  B  e.  V  /\  C  e.  W )  ->  [_ C  /  x ]_ B  e. 
_V )
20 ssexg 3999 . 2  |-  ( ( ( F `  C
)  C_  [_ C  /  x ]_ B  /\  [_ C  /  x ]_ B  e. 
_V )  ->  ( F `  C )  e.  _V )
218, 19, 20sylancr 406 1  |-  ( ( A. x  B  e.  V  /\  C  e.  W )  ->  ( F `  C )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1294    = wceq 1296    e. wcel 1445   _Vcvv 2633   [_csb 2947    C_ wss 3013    |-> cmpt 3921   ` cfv 5049
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-iota 5014  df-fun 5051  df-fv 5057
This theorem is referenced by:  mpt2fvex  6011  xpcomco  6622
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