ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpt2fvex Unicode version

Theorem mpt2fvex 5955
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 5637 . 2  |-  ( R F S )  =  ( F `  <. R ,  S >. )
2 elex 2630 . . . . . . . . 9  |-  ( C  e.  V  ->  C  e.  _V )
32alimi 1389 . . . . . . . 8  |-  ( A. y  C  e.  V  ->  A. y  C  e. 
_V )
4 vex 2622 . . . . . . . . 9  |-  z  e. 
_V
5 2ndexg 5921 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 2nd `  z )  e. 
_V )
6 nfcv 2228 . . . . . . . . . 10  |-  F/_ y
( 2nd `  z
)
7 nfcsb1v 2961 . . . . . . . . . . 11  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ C
87nfel1 2239 . . . . . . . . . 10  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ C  e.  _V
9 csbeq1a 2939 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  z
)  ->  C  =  [_ ( 2nd `  z
)  /  y ]_ C )
109eleq1d 2156 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( C  e.  _V  <->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
116, 8, 10spcgf 2701 . . . . . . . . 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 1389 . . . . . 6  |-  ( A. x A. y  C  e.  V  ->  A. x [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
15 1stexg 5920 . . . . . . 7  |-  ( z  e.  _V  ->  ( 1st `  z )  e. 
_V )
16 nfcv 2228 . . . . . . . 8  |-  F/_ x
( 1st `  z
)
17 nfcsb1v 2961 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C
1817nfel1 2239 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V
19 csbeq1a 2939 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ C  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
2019eleq1d 2156 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( [_ ( 2nd `  z )  /  y ]_ C  e.  _V  <->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
2116, 18, 20spcgf 2701 . . . . . . 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 1802 . . . 4  |-  ( A. x A. y  C  e.  V  ->  A. z [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
25243ad2ant1 964 . . 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 4046 . . . 4  |-  ( ( R  e.  W  /\  S  e.  X )  -> 
<. R ,  S >.  e. 
_V )
27263adant1 961 . . 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 5949 . . . . 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 2108 . . . 4  |-  F  =  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
3130mptfvex 5372 . . 3  |-  ( ( A. z [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V  /\ 
<. R ,  S >.  e. 
_V )  ->  ( F `  <. R ,  S >. )  e.  _V )
3225, 27, 31syl2anc 403 . 2  |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X )  ->  ( F `  <. R ,  S >. )  e.  _V )
331, 32syl5eqel 2174 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 924   A.wal 1287    = wceq 1289    e. wcel 1438   _Vcvv 2619   [_csb 2931   {csn 3441   <.cop 3444   U_ciun 3725    |-> cmpt 3891    X. cxp 4426   ` cfv 5002  (class class class)co 5634    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894
This theorem is referenced by:  mpt2fvexi  5958  oaexg  6191  omexg  6194  oeiexg  6196
  Copyright terms: Public domain W3C validator