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Theorem prismorcset2 25918
Description: The predicate "is a morphism of the category Set". (Contributed by FL, 15-Sep-2013.)
Hypotheses
Ref Expression
prismorcset2.1  |-  A  =  ( ( 1st  o.  1st ) `  M )
prismorcset2.2  |-  B  =  ( ( 2nd  o.  1st ) `  M )
prismorcset2.3  |-  C  =  ( 2nd `  M
)
Assertion
Ref Expression
prismorcset2  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( A  e.  U  /\  B  e.  U  /\  C  e.  ( B  ^m  A
) ) )

Proof of Theorem prismorcset2
Dummy variables  m  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prismorcset2.1 . 2  |-  A  =  ( ( 1st  o.  1st ) `  M )
2 prismorcset2.2 . . . 4  |-  B  =  ( ( 2nd  o.  1st ) `  M )
3 prismorcset2.3 . . . . . 6  |-  C  =  ( 2nd `  M
)
4 fveq2 5525 . . . . . . . . . . . 12  |-  ( m  =  M  ->  (
( 1st  o.  1st ) `  m )  =  ( ( 1st 
o.  1st ) `  M
) )
54eleq1d 2349 . . . . . . . . . . 11  |-  ( m  =  M  ->  (
( ( 1st  o.  1st ) `  m )  e.  U  <->  ( ( 1st  o.  1st ) `  M )  e.  U
) )
6 fveq2 5525 . . . . . . . . . . . 12  |-  ( m  =  M  ->  (
( 2nd  o.  1st ) `  m )  =  ( ( 2nd 
o.  1st ) `  M
) )
76eleq1d 2349 . . . . . . . . . . 11  |-  ( m  =  M  ->  (
( ( 2nd  o.  1st ) `  m )  e.  U  <->  ( ( 2nd  o.  1st ) `  M )  e.  U
) )
8 fveq2 5525 . . . . . . . . . . . 12  |-  ( m  =  M  ->  ( 2nd `  m )  =  ( 2nd `  M
) )
96, 4oveq12d 5876 . . . . . . . . . . . 12  |-  ( m  =  M  ->  (
( ( 2nd  o.  1st ) `  m )  ^m  ( ( 1st 
o.  1st ) `  m
) )  =  ( ( ( 2nd  o.  1st ) `  M )  ^m  ( ( 1st 
o.  1st ) `  M
) ) )
108, 9eleq12d 2351 . . . . . . . . . . 11  |-  ( m  =  M  ->  (
( 2nd `  m
)  e.  ( ( ( 2nd  o.  1st ) `  m )  ^m  ( ( 1st  o.  1st ) `  m ) )  <->  ( 2nd `  M
)  e.  ( ( ( 2nd  o.  1st ) `  M )  ^m  ( ( 1st  o.  1st ) `  M ) ) ) )
115, 7, 103anbi123d 1252 . . . . . . . . . 10  |-  ( m  =  M  ->  (
( ( ( 1st 
o.  1st ) `  m
)  e.  U  /\  ( ( 2nd  o.  1st ) `  m )  e.  U  /\  ( 2nd `  m )  e.  ( ( ( 2nd 
o.  1st ) `  m
)  ^m  ( ( 1st  o.  1st ) `  m ) ) )  <-> 
( ( ( 1st 
o.  1st ) `  M
)  e.  U  /\  ( ( 2nd  o.  1st ) `  M )  e.  U  /\  ( 2nd `  M )  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) ) ) )
1211imbi2d 307 . . . . . . . . 9  |-  ( m  =  M  ->  (
( U  e.  Univ  -> 
( ( ( 1st 
o.  1st ) `  m
)  e.  U  /\  ( ( 2nd  o.  1st ) `  m )  e.  U  /\  ( 2nd `  m )  e.  ( ( ( 2nd 
o.  1st ) `  m
)  ^m  ( ( 1st  o.  1st ) `  m ) ) ) )  <->  ( U  e. 
Univ  ->  ( ( ( 1st  o.  1st ) `  M )  e.  U  /\  ( ( 2nd  o.  1st ) `  M )  e.  U  /\  ( 2nd `  M )  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) ) ) ) )
13 morcatset1 25915 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  ( Morphism SetCat `  U )  =  { <. <. a ,  b
>. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  (
b  ^m  a )
) } )
1413eleq2d 2350 . . . . . . . . . . 11  |-  ( U  e.  Univ  ->  ( m  e.  ( Morphism SetCat `  U
)  <->  m  e.  { <. <.
a ,  b >. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a ) ) } ) )
15 eleq1 2343 . . . . . . . . . . . . 13  |-  ( a  =  ( 1st `  ( 1st `  m ) )  ->  ( a  e.  U  <->  ( 1st `  ( 1st `  m ) )  e.  U ) )
16 oveq2 5866 . . . . . . . . . . . . . 14  |-  ( a  =  ( 1st `  ( 1st `  m ) )  ->  ( b  ^m  a )  =  ( b  ^m  ( 1st `  ( 1st `  m
) ) ) )
1716eleq2d 2350 . . . . . . . . . . . . 13  |-  ( a  =  ( 1st `  ( 1st `  m ) )  ->  ( c  e.  ( b  ^m  a
)  <->  c  e.  ( b  ^m  ( 1st `  ( 1st `  m
) ) ) ) )
1815, 173anbi13d 1254 . . . . . . . . . . . 12  |-  ( a  =  ( 1st `  ( 1st `  m ) )  ->  ( ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a
) )  <->  ( ( 1st `  ( 1st `  m
) )  e.  U  /\  b  e.  U  /\  c  e.  (
b  ^m  ( 1st `  ( 1st `  m
) ) ) ) ) )
19 eleq1 2343 . . . . . . . . . . . . 13  |-  ( b  =  ( 2nd `  ( 1st `  m ) )  ->  ( b  e.  U  <->  ( 2nd `  ( 1st `  m ) )  e.  U ) )
20 oveq1 5865 . . . . . . . . . . . . . 14  |-  ( b  =  ( 2nd `  ( 1st `  m ) )  ->  ( b  ^m  ( 1st `  ( 1st `  m ) ) )  =  ( ( 2nd `  ( 1st `  m
) )  ^m  ( 1st `  ( 1st `  m
) ) ) )
2120eleq2d 2350 . . . . . . . . . . . . 13  |-  ( b  =  ( 2nd `  ( 1st `  m ) )  ->  ( c  e.  ( b  ^m  ( 1st `  ( 1st `  m
) ) )  <->  c  e.  ( ( 2nd `  ( 1st `  m ) )  ^m  ( 1st `  ( 1st `  m ) ) ) ) )
2219, 213anbi23d 1255 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  ( 1st `  m ) )  ->  ( ( ( 1st `  ( 1st `  m ) )  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  ( 1st `  ( 1st `  m
) ) ) )  <-> 
( ( 1st `  ( 1st `  m ) )  e.  U  /\  ( 2nd `  ( 1st `  m
) )  e.  U  /\  c  e.  (
( 2nd `  ( 1st `  m ) )  ^m  ( 1st `  ( 1st `  m ) ) ) ) ) )
23 fo1st 6139 . . . . . . . . . . . . . . . . . 18  |-  1st : _V -onto-> _V
24 fof 5451 . . . . . . . . . . . . . . . . . 18  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
2523, 24ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  1st : _V
--> _V
26 vex 2791 . . . . . . . . . . . . . . . . 17  |-  m  e. 
_V
27 fvco3 5596 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st : _V --> _V  /\  m  e.  _V )  ->  ( ( 1st  o.  1st ) `  m )  =  ( 1st `  ( 1st `  m ) ) )
2825, 26, 27mp2an 653 . . . . . . . . . . . . . . . 16  |-  ( ( 1st  o.  1st ) `  m )  =  ( 1st `  ( 1st `  m ) )
2928eqcomi 2287 . . . . . . . . . . . . . . 15  |-  ( 1st `  ( 1st `  m
) )  =  ( ( 1st  o.  1st ) `  m )
3029eleq1i 2346 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( 1st `  m ) )  e.  U  <->  ( ( 1st 
o.  1st ) `  m
)  e.  U )
3130a1i 10 . . . . . . . . . . . . 13  |-  ( c  =  ( 2nd `  m
)  ->  ( ( 1st `  ( 1st `  m
) )  e.  U  <->  ( ( 1st  o.  1st ) `  m )  e.  U ) )
32 fvco3 5596 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st : _V --> _V  /\  m  e.  _V )  ->  ( ( 2nd  o.  1st ) `  m )  =  ( 2nd `  ( 1st `  m ) ) )
3325, 26, 32mp2an 653 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd  o.  1st ) `  m )  =  ( 2nd `  ( 1st `  m ) )
3433eqcomi 2287 . . . . . . . . . . . . . . 15  |-  ( 2nd `  ( 1st `  m
) )  =  ( ( 2nd  o.  1st ) `  m )
3534eleq1i 2346 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  ( 1st `  m ) )  e.  U  <->  ( ( 2nd 
o.  1st ) `  m
)  e.  U )
3635a1i 10 . . . . . . . . . . . . 13  |-  ( c  =  ( 2nd `  m
)  ->  ( ( 2nd `  ( 1st `  m
) )  e.  U  <->  ( ( 2nd  o.  1st ) `  m )  e.  U ) )
37 id 19 . . . . . . . . . . . . . 14  |-  ( c  =  ( 2nd `  m
)  ->  c  =  ( 2nd `  m ) )
3834, 29oveq12i 5870 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  m ) )  ^m  ( 1st `  ( 1st `  m ) ) )  =  ( ( ( 2nd  o.  1st ) `  m )  ^m  (
( 1st  o.  1st ) `  m )
)
3938a1i 10 . . . . . . . . . . . . . 14  |-  ( c  =  ( 2nd `  m
)  ->  ( ( 2nd `  ( 1st `  m
) )  ^m  ( 1st `  ( 1st `  m
) ) )  =  ( ( ( 2nd 
o.  1st ) `  m
)  ^m  ( ( 1st  o.  1st ) `  m ) ) )
4037, 39eleq12d 2351 . . . . . . . . . . . . 13  |-  ( c  =  ( 2nd `  m
)  ->  ( c  e.  ( ( 2nd `  ( 1st `  m ) )  ^m  ( 1st `  ( 1st `  m ) ) )  <->  ( 2nd `  m
)  e.  ( ( ( 2nd  o.  1st ) `  m )  ^m  ( ( 1st  o.  1st ) `  m ) ) ) )
4131, 36, 403anbi123d 1252 . . . . . . . . . . . 12  |-  ( c  =  ( 2nd `  m
)  ->  ( (
( 1st `  ( 1st `  m ) )  e.  U  /\  ( 2nd `  ( 1st `  m
) )  e.  U  /\  c  e.  (
( 2nd `  ( 1st `  m ) )  ^m  ( 1st `  ( 1st `  m ) ) ) )  <->  ( (
( 1st  o.  1st ) `  m )  e.  U  /\  (
( 2nd  o.  1st ) `  m )  e.  U  /\  ( 2nd `  m )  e.  ( ( ( 2nd 
o.  1st ) `  m
)  ^m  ( ( 1st  o.  1st ) `  m ) ) ) ) )
4218, 22, 41eloprabi 6186 . . . . . . . . . . 11  |-  ( m  e.  { <. <. a ,  b >. ,  c
>.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a
) ) }  ->  ( ( ( 1st  o.  1st ) `  m )  e.  U  /\  (
( 2nd  o.  1st ) `  m )  e.  U  /\  ( 2nd `  m )  e.  ( ( ( 2nd 
o.  1st ) `  m
)  ^m  ( ( 1st  o.  1st ) `  m ) ) ) )
4314, 42syl6bi 219 . . . . . . . . . 10  |-  ( U  e.  Univ  ->  ( m  e.  ( Morphism SetCat `  U
)  ->  ( (
( 1st  o.  1st ) `  m )  e.  U  /\  (
( 2nd  o.  1st ) `  m )  e.  U  /\  ( 2nd `  m )  e.  ( ( ( 2nd 
o.  1st ) `  m
)  ^m  ( ( 1st  o.  1st ) `  m ) ) ) ) )
4443com12 27 . . . . . . . . 9  |-  ( m  e.  ( Morphism SetCat `  U
)  ->  ( U  e.  Univ  ->  ( (
( 1st  o.  1st ) `  m )  e.  U  /\  (
( 2nd  o.  1st ) `  m )  e.  U  /\  ( 2nd `  m )  e.  ( ( ( 2nd 
o.  1st ) `  m
)  ^m  ( ( 1st  o.  1st ) `  m ) ) ) ) )
4512, 44vtoclga 2849 . . . . . . . 8  |-  ( M  e.  ( Morphism SetCat `  U
)  ->  ( U  e.  Univ  ->  ( (
( 1st  o.  1st ) `  M )  e.  U  /\  (
( 2nd  o.  1st ) `  M )  e.  U  /\  ( 2nd `  M )  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) ) ) )
4645impcom 419 . . . . . . 7  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( (
( 1st  o.  1st ) `  M )  e.  U  /\  (
( 2nd  o.  1st ) `  M )  e.  U  /\  ( 2nd `  M )  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) ) )
47 eleq1 2343 . . . . . . . 8  |-  ( C  =  ( 2nd `  M
)  ->  ( C  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) )  <->  ( 2nd `  M )  e.  ( ( ( 2nd  o.  1st ) `  M )  ^m  ( ( 1st 
o.  1st ) `  M
) ) ) )
48473anbi3d 1258 . . . . . . 7  |-  ( C  =  ( 2nd `  M
)  ->  ( (
( ( 1st  o.  1st ) `  M )  e.  U  /\  (
( 2nd  o.  1st ) `  M )  e.  U  /\  C  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) )  <-> 
( ( ( 1st 
o.  1st ) `  M
)  e.  U  /\  ( ( 2nd  o.  1st ) `  M )  e.  U  /\  ( 2nd `  M )  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) ) ) )
4946, 48syl5ibr 212 . . . . . 6  |-  ( C  =  ( 2nd `  M
)  ->  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( (
( 1st  o.  1st ) `  M )  e.  U  /\  (
( 2nd  o.  1st ) `  M )  e.  U  /\  C  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) ) ) )
503, 49ax-mp 8 . . . . 5  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( (
( 1st  o.  1st ) `  M )  e.  U  /\  (
( 2nd  o.  1st ) `  M )  e.  U  /\  C  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) ) )
51 eleq1 2343 . . . . . 6  |-  ( B  =  ( ( 2nd 
o.  1st ) `  M
)  ->  ( B  e.  U  <->  ( ( 2nd 
o.  1st ) `  M
)  e.  U ) )
52 oveq1 5865 . . . . . . 7  |-  ( B  =  ( ( 2nd 
o.  1st ) `  M
)  ->  ( B  ^m  ( ( 1st  o.  1st ) `  M ) )  =  ( ( ( 2nd  o.  1st ) `  M )  ^m  ( ( 1st  o.  1st ) `  M ) ) )
5352eleq2d 2350 . . . . . 6  |-  ( B  =  ( ( 2nd 
o.  1st ) `  M
)  ->  ( C  e.  ( B  ^m  (
( 1st  o.  1st ) `  M )
)  <->  C  e.  (
( ( 2nd  o.  1st ) `  M )  ^m  ( ( 1st 
o.  1st ) `  M
) ) ) )
5451, 533anbi23d 1255 . . . . 5  |-  ( B  =  ( ( 2nd 
o.  1st ) `  M
)  ->  ( (
( ( 1st  o.  1st ) `  M )  e.  U  /\  B  e.  U  /\  C  e.  ( B  ^m  (
( 1st  o.  1st ) `  M )
) )  <->  ( (
( 1st  o.  1st ) `  M )  e.  U  /\  (
( 2nd  o.  1st ) `  M )  e.  U  /\  C  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) ) ) )
5550, 54syl5ibr 212 . . . 4  |-  ( B  =  ( ( 2nd 
o.  1st ) `  M
)  ->  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( (
( 1st  o.  1st ) `  M )  e.  U  /\  B  e.  U  /\  C  e.  ( B  ^m  (
( 1st  o.  1st ) `  M )
) ) ) )
562, 55ax-mp 8 . . 3  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( (
( 1st  o.  1st ) `  M )  e.  U  /\  B  e.  U  /\  C  e.  ( B  ^m  (
( 1st  o.  1st ) `  M )
) ) )
57 eleq1 2343 . . . 4  |-  ( A  =  ( ( 1st 
o.  1st ) `  M
)  ->  ( A  e.  U  <->  ( ( 1st 
o.  1st ) `  M
)  e.  U ) )
58 oveq2 5866 . . . . 5  |-  ( A  =  ( ( 1st 
o.  1st ) `  M
)  ->  ( B  ^m  A )  =  ( B  ^m  ( ( 1st  o.  1st ) `  M ) ) )
5958eleq2d 2350 . . . 4  |-  ( A  =  ( ( 1st 
o.  1st ) `  M
)  ->  ( C  e.  ( B  ^m  A
)  <->  C  e.  ( B  ^m  ( ( 1st 
o.  1st ) `  M
) ) ) )
6057, 593anbi13d 1254 . . 3  |-  ( A  =  ( ( 1st 
o.  1st ) `  M
)  ->  ( ( A  e.  U  /\  B  e.  U  /\  C  e.  ( B  ^m  A ) )  <->  ( (
( 1st  o.  1st ) `  M )  e.  U  /\  B  e.  U  /\  C  e.  ( B  ^m  (
( 1st  o.  1st ) `  M )
) ) ) )
6156, 60syl5ibr 212 . 2  |-  ( A  =  ( ( 1st 
o.  1st ) `  M
)  ->  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( A  e.  U  /\  B  e.  U  /\  C  e.  ( B  ^m  A
) ) ) )
621, 61ax-mp 8 1  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( A  e.  U  /\  B  e.  U  /\  C  e.  ( B  ^m  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   {coprab 5859   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   Univcgru 8412   Morphism SetCatccmrcase 25910
This theorem is referenced by:  prismorcset3  25938  rocatval  25959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-1st 6122  df-2nd 6123  df-morcatset 25911
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