MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sscfn2 Structured version   Unicode version

Theorem sscfn2 14023
Description: The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
sscfn1.1  |-  ( ph  ->  H  C_cat  J )
sscfn2.2  |-  ( ph  ->  T  =  dom  dom  J )
Assertion
Ref Expression
sscfn2  |-  ( ph  ->  J  Fn  ( T  X.  T ) )

Proof of Theorem sscfn2
Dummy variables  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sscfn1.1 . . 3  |-  ( ph  ->  H  C_cat  J )
2 brssc 14019 . . 3  |-  ( H 
C_cat  J  <->  E. t ( J  Fn  ( t  X.  t )  /\  E. y  e.  ~P  t H  e.  X_ x  e.  ( y  X.  y
) ~P ( J `
 x ) ) )
31, 2sylib 190 . 2  |-  ( ph  ->  E. t ( J  Fn  ( t  X.  t )  /\  E. y  e.  ~P  t H  e.  X_ x  e.  ( y  X.  y
) ~P ( J `
 x ) ) )
4 simpr 449 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( t  X.  t
) )
5 sscfn2.2 . . . . . . . . . 10  |-  ( ph  ->  T  =  dom  dom  J )
65adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  T  =  dom  dom  J )
7 fndm 5547 . . . . . . . . . . . 12  |-  ( J  Fn  ( t  X.  t )  ->  dom  J  =  ( t  X.  t ) )
87adantl 454 . . . . . . . . . . 11  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  J  =  ( t  X.  t ) )
98dmeqd 5075 . . . . . . . . . 10  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  dom  (
t  X.  t ) )
10 dmxpid 5092 . . . . . . . . . 10  |-  dom  (
t  X.  t )  =  t
119, 10syl6eq 2486 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  t )
126, 11eqtr2d 2471 . . . . . . . 8  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  t  =  T )
1312, 12xpeq12d 4906 . . . . . . 7  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  (
t  X.  t )  =  ( T  X.  T ) )
1413fneq2d 5540 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  ( J  Fn  ( t  X.  t )  <->  J  Fn  ( T  X.  T
) ) )
154, 14mpbid 203 . . . . 5  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( T  X.  T
) )
1615ex 425 . . . 4  |-  ( ph  ->  ( J  Fn  (
t  X.  t )  ->  J  Fn  ( T  X.  T ) ) )
1716adantrd 456 . . 3  |-  ( ph  ->  ( ( J  Fn  ( t  X.  t
)  /\  E. y  e.  ~P  t H  e.  X_ x  e.  (
y  X.  y ) ~P ( J `  x ) )  ->  J  Fn  ( T  X.  T ) ) )
1817exlimdv 1647 . 2  |-  ( ph  ->  ( E. t ( J  Fn  ( t  X.  t )  /\  E. y  e.  ~P  t H  e.  X_ x  e.  ( y  X.  y
) ~P ( J `
 x ) )  ->  J  Fn  ( T  X.  T ) ) )
193, 18mpd 15 1  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2708   ~Pcpw 3801   class class class wbr 4215    X. cxp 4879   dom cdm 4881    Fn wfn 5452   ` cfv 5457   X_cixp 7066    C_cat cssc 14012
This theorem is referenced by:  ssc2  14027  ssctr  14030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ixp 7067  df-ssc 14015
  Copyright terms: Public domain W3C validator