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Theorem sscfn1 13694
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 )
sscfn1.2  |-  ( ph  ->  S  =  dom  dom  H )
Assertion
Ref Expression
sscfn1  |-  ( ph  ->  H  Fn  ( S  X.  S ) )

Proof of Theorem sscfn1
Dummy variables  t 
s  x 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. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x ) ) )
31, 2sylib 188 . 2  |-  ( ph  ->  E. t ( J  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x ) ) )
4 ixpfn 6822 . . . . . 6  |-  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( s  X.  s
) )
5 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  H  Fn  ( s  X.  s
) )
6 sscfn1.2 . . . . . . . . . . . 12  |-  ( ph  ->  S  =  dom  dom  H )
76adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  S  =  dom  dom  H )
8 fndm 5343 . . . . . . . . . . . . . 14  |-  ( H  Fn  ( s  X.  s )  ->  dom  H  =  ( s  X.  s ) )
98adantl 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  H  =  ( s  X.  s ) )
109dmeqd 4881 . . . . . . . . . . . 12  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  dom  (
s  X.  s ) )
11 dmxpid 4898 . . . . . . . . . . . 12  |-  dom  (
s  X.  s )  =  s
1210, 11syl6eq 2331 . . . . . . . . . . 11  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  s )
137, 12eqtr2d 2316 . . . . . . . . . 10  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  s  =  S )
1413, 13xpeq12d 4714 . . . . . . . . 9  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  (
s  X.  s )  =  ( S  X.  S ) )
1514fneq2d 5336 . . . . . . . 8  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  ( H  Fn  ( s  X.  s )  <->  H  Fn  ( S  X.  S
) ) )
165, 15mpbid 201 . . . . . . 7  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  H  Fn  ( S  X.  S
) )
1716ex 423 . . . . . 6  |-  ( ph  ->  ( H  Fn  (
s  X.  s )  ->  H  Fn  ( S  X.  S ) ) )
184, 17syl5 28 . . . . 5  |-  ( ph  ->  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( S  X.  S ) ) )
1918rexlimdvw 2670 . . . 4  |-  ( ph  ->  ( E. s  e. 
~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( S  X.  S
) ) )
2019adantld 453 . . 3  |-  ( ph  ->  ( ( J  Fn  ( t  X.  t
)  /\  E. s  e.  ~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x ) )  ->  H  Fn  ( S  X.  S ) ) )
2120exlimdv 1664 . 2  |-  ( ph  ->  ( E. t ( J  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x ) )  ->  H  Fn  ( S  X.  S ) ) )
223, 21mpd 14 1  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
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:  ssctr  13702  ssceq  13703  subcfn  13715  subsubc  13727
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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