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Theorem sscfn2 13695
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 13691 . . 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 188 . 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 447 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( t  X.  t
) )
5 sscfn2.2 . . . . . . . . . 10  |-  ( ph  ->  T  =  dom  dom  J )
65adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  T  =  dom  dom  J )
7 fndm 5343 . . . . . . . . . . . 12  |-  ( J  Fn  ( t  X.  t )  ->  dom  J  =  ( t  X.  t ) )
84, 7syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  J  =  ( t  X.  t ) )
98dmeqd 4881 . . . . . . . . . 10  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  dom  (
t  X.  t ) )
10 dmxpid 4898 . . . . . . . . . 10  |-  dom  (
t  X.  t )  =  t
119, 10syl6eq 2331 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  t )
126, 11eqtr2d 2316 . . . . . . . 8  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  t  =  T )
1312, 12xpeq12d 4714 . . . . . . 7  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  (
t  X.  t )  =  ( T  X.  T ) )
1413fneq2d 5336 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  ( J  Fn  ( t  X.  t )  <->  J  Fn  ( T  X.  T
) ) )
154, 14mpbid 201 . . . . 5  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( T  X.  T
) )
1615ex 423 . . . 4  |-  ( ph  ->  ( J  Fn  (
t  X.  t )  ->  J  Fn  ( T  X.  T ) ) )
1716adantrd 454 . . 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 1664 . 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 14 1  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   ~Pcpw 3625   class class class wbr 4023    X. cxp 4687   dom cdm 4689    Fn wfn 5250   ` cfv 5255   X_cixp 6817    C_cat cssc 13684
This theorem is referenced by:  ssc2  13699  ssctr  13702
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-ixp 6818  df-ssc 13687
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