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Theorem mpt2fvex 5829
Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypothesis
Ref Expression
fmpt2.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
mpt2fvex  |-  ( ( 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 mpt2fvex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-ov 5515 . 2  |-  ( R F S )  =  ( F `  <. R ,  S >. )
2 elex 2566 . . . . . . . . 9  |-  ( C  e.  V  ->  C  e.  _V )
32alimi 1344 . . . . . . . 8  |-  ( A. y  C  e.  V  ->  A. y  C  e. 
_V )
4 vex 2560 . . . . . . . . 9  |-  z  e. 
_V
5 2ndexg 5795 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 2nd `  z )  e. 
_V )
6 nfcv 2178 . . . . . . . . . 10  |-  F/_ y
( 2nd `  z
)
7 nfcsb1v 2882 . . . . . . . . . . 11  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ C
87nfel1 2188 . . . . . . . . . 10  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ C  e.  _V
9 csbeq1a 2860 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  z
)  ->  C  =  [_ ( 2nd `  z
)  /  y ]_ C )
109eleq1d 2106 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( C  e.  _V  <->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
116, 8, 10spcgf 2635 . . . . . . . . 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 1344 . . . . . 6  |-  ( A. x A. y  C  e.  V  ->  A. x [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
15 1stexg 5794 . . . . . . 7  |-  ( z  e.  _V  ->  ( 1st `  z )  e. 
_V )
16 nfcv 2178 . . . . . . . 8  |-  F/_ x
( 1st `  z
)
17 nfcsb1v 2882 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C
1817nfel1 2188 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V
19 csbeq1a 2860 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ C  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
2019eleq1d 2106 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( [_ ( 2nd `  z )  /  y ]_ C  e.  _V  <->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
2116, 18, 20spcgf 2635 . . . . . . 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 1754 . . . 4  |-  ( A. x A. y  C  e.  V  ->  A. z [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
25243ad2ant1 925 . . 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 3964 . . . 4  |-  ( ( R  e.  W  /\  S  e.  X )  -> 
<. R ,  S >.  e. 
_V )
27263adant1 922 . . 3  |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X )  -> 
<. R ,  S >.  e. 
_V )
28 fmpt2.1 . . . . 5  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
29 mpt2mptsx 5823 . . . . 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 2060 . . . 4  |-  F  =  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
3130mptfvex 5256 . . 3  |-  ( ( A. z [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V  /\ 
<. R ,  S >.  e. 
_V )  ->  ( F `  <. R ,  S >. )  e.  _V )
3225, 27, 31syl2anc 391 . 2  |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X )  ->  ( F `  <. R ,  S >. )  e.  _V )
331, 32syl5eqel 2124 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 885   A.wal 1241    = wceq 1243    e. wcel 1393   _Vcvv 2557   [_csb 2852   {csn 3375   <.cop 3378   U_ciun 3657    |-> cmpt 3818    X. cxp 4343   ` cfv 4902  (class class class)co 5512    |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768
This theorem is referenced by:  mpt2fvexi  5832  oaexg  6028  omexg  6031  oeiexg  6033
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