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Theorem mpofvex 6108
Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypothesis
Ref Expression
fmpo.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
mpofvex  |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X )  ->  ( R F S )  e.  _V )
Distinct variable groups:    x, A, y   
x, B, y
Allowed substitution hints:    C( x, y)    R( x, y)    S( x, y)    F( x, y)    V( x, y)    W( x, y)    X( x, y)

Proof of Theorem mpofvex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-ov 5784 . 2  |-  ( R F S )  =  ( F `  <. R ,  S >. )
2 elex 2700 . . . . . . . . 9  |-  ( C  e.  V  ->  C  e.  _V )
32alimi 1432 . . . . . . . 8  |-  ( A. y  C  e.  V  ->  A. y  C  e. 
_V )
4 vex 2692 . . . . . . . . 9  |-  z  e. 
_V
5 2ndexg 6073 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 2nd `  z )  e. 
_V )
6 nfcv 2282 . . . . . . . . . 10  |-  F/_ y
( 2nd `  z
)
7 nfcsb1v 3039 . . . . . . . . . . 11  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ C
87nfel1 2293 . . . . . . . . . 10  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ C  e.  _V
9 csbeq1a 3015 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  z
)  ->  C  =  [_ ( 2nd `  z
)  /  y ]_ C )
109eleq1d 2209 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( C  e.  _V  <->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
116, 8, 10spcgf 2771 . . . . . . . . 9  |-  ( ( 2nd `  z )  e.  _V  ->  ( A. y  C  e.  _V  ->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
124, 5, 11mp2b 8 . . . . . . . 8  |-  ( A. y  C  e.  _V  ->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
133, 12syl 14 . . . . . . 7  |-  ( A. y  C  e.  V  ->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
1413alimi 1432 . . . . . 6  |-  ( A. x A. y  C  e.  V  ->  A. x [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
15 1stexg 6072 . . . . . . 7  |-  ( z  e.  _V  ->  ( 1st `  z )  e. 
_V )
16 nfcv 2282 . . . . . . . 8  |-  F/_ x
( 1st `  z
)
17 nfcsb1v 3039 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C
1817nfel1 2293 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V
19 csbeq1a 3015 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ C  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
2019eleq1d 2209 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( [_ ( 2nd `  z )  /  y ]_ C  e.  _V  <->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
2116, 18, 20spcgf 2771 . . . . . . 7  |-  ( ( 1st `  z )  e.  _V  ->  ( A. x [_ ( 2nd `  z )  /  y ]_ C  e.  _V  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
224, 15, 21mp2b 8 . . . . . 6  |-  ( A. x [_ ( 2nd `  z
)  /  y ]_ C  e.  _V  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
2314, 22syl 14 . . . . 5  |-  ( A. x A. y  C  e.  V  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
2423alrimiv 1847 . . . 4  |-  ( A. x A. y  C  e.  V  ->  A. z [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
25243ad2ant1 1003 . . 3  |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X )  ->  A. z [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V )
26 opexg 4157 . . . 4  |-  ( ( R  e.  W  /\  S  e.  X )  -> 
<. R ,  S >.  e. 
_V )
27263adant1 1000 . . 3  |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X )  -> 
<. R ,  S >.  e. 
_V )
28 fmpo.1 . . . . 5  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
29 mpomptsx 6102 . . . . 5  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
3028, 29eqtri 2161 . . . 4  |-  F  =  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
3130mptfvex 5513 . . 3  |-  ( ( A. z [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V  /\ 
<. R ,  S >.  e. 
_V )  ->  ( F `  <. R ,  S >. )  e.  _V )
3225, 27, 31syl2anc 409 . 2  |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X )  ->  ( F `  <. R ,  S >. )  e.  _V )
331, 32eqeltrid 2227 1  |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X )  ->  ( R F S )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 963   A.wal 1330    = wceq 1332    e. wcel 1481   _Vcvv 2689   [_csb 3006   {csn 3531   <.cop 3534   U_ciun 3820    |-> cmpt 3996    X. cxp 4544   ` cfv 5130  (class class class)co 5781    e. cmpo 5783   1stc1st 6043   2ndc2nd 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fo 5136  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046
This theorem is referenced by:  mpofvexi  6111  oaexg  6351  omexg  6354  oeiexg  6356
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