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Theorem mpt2fvex 5857
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 5543 . 2  |-  ( R F S )  =  ( F `  <. R ,  S >. )
2 elex 2583 . . . . . . . . 9  |-  ( C  e.  V  ->  C  e.  _V )
32alimi 1360 . . . . . . . 8  |-  ( A. y  C  e.  V  ->  A. y  C  e. 
_V )
4 vex 2577 . . . . . . . . 9  |-  z  e. 
_V
5 2ndexg 5823 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 2nd `  z )  e. 
_V )
6 nfcv 2194 . . . . . . . . . 10  |-  F/_ y
( 2nd `  z
)
7 nfcsb1v 2910 . . . . . . . . . . 11  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ C
87nfel1 2204 . . . . . . . . . 10  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ C  e.  _V
9 csbeq1a 2888 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  z
)  ->  C  =  [_ ( 2nd `  z
)  /  y ]_ C )
109eleq1d 2122 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( C  e.  _V  <->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
116, 8, 10spcgf 2652 . . . . . . . . 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 1360 . . . . . 6  |-  ( A. x A. y  C  e.  V  ->  A. x [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
15 1stexg 5822 . . . . . . 7  |-  ( z  e.  _V  ->  ( 1st `  z )  e. 
_V )
16 nfcv 2194 . . . . . . . 8  |-  F/_ x
( 1st `  z
)
17 nfcsb1v 2910 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C
1817nfel1 2204 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V
19 csbeq1a 2888 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ C  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
2019eleq1d 2122 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( [_ ( 2nd `  z )  /  y ]_ C  e.  _V  <->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
2116, 18, 20spcgf 2652 . . . . . . 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 1770 . . . 4  |-  ( A. x A. y  C  e.  V  ->  A. z [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
25243ad2ant1 936 . . 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 3992 . . . 4  |-  ( ( R  e.  W  /\  S  e.  X )  -> 
<. R ,  S >.  e. 
_V )
27263adant1 933 . . 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 5851 . . . . 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 2076 . . . 4  |-  F  =  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
3130mptfvex 5284 . . 3  |-  ( ( A. z [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V  /\ 
<. R ,  S >.  e. 
_V )  ->  ( F `  <. R ,  S >. )  e.  _V )
3225, 27, 31syl2anc 397 . 2  |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X )  ->  ( F `  <. R ,  S >. )  e.  _V )
331, 32syl5eqel 2140 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 896   A.wal 1257    = wceq 1259    e. wcel 1409   _Vcvv 2574   [_csb 2880   {csn 3403   <.cop 3406   U_ciun 3685    |-> cmpt 3846    X. cxp 4371   ` cfv 4930  (class class class)co 5540    |-> cmpt2 5542   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796
This theorem is referenced by:  mpt2fvexi  5860  oaexg  6059  omexg  6062  oeiexg  6064
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