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Theorem sscfn1 15246
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 15243 . . 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 196 . 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 7416 . . . . . 6  |-  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( s  X.  s
) )
5 simpr 459 . . . . . . . 8  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  H  Fn  ( s  X.  s
) )
6 sscfn1.2 . . . . . . . . . . . 12  |-  ( ph  ->  S  =  dom  dom  H )
76adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  S  =  dom  dom  H )
8 fndm 5605 . . . . . . . . . . . . . 14  |-  ( H  Fn  ( s  X.  s )  ->  dom  H  =  ( s  X.  s ) )
98adantl 464 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  H  =  ( s  X.  s ) )
109dmeqd 5135 . . . . . . . . . . . 12  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  dom  (
s  X.  s ) )
11 dmxpid 5152 . . . . . . . . . . . 12  |-  dom  (
s  X.  s )  =  s
1210, 11syl6eq 2453 . . . . . . . . . . 11  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  s )
137, 12eqtr2d 2438 . . . . . . . . . 10  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  s  =  S )
1413sqxpeqd 4956 . . . . . . . . 9  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  (
s  X.  s )  =  ( S  X.  S ) )
1514fneq2d 5597 . . . . . . . 8  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  ( H  Fn  ( s  X.  s )  <->  H  Fn  ( S  X.  S
) ) )
165, 15mpbid 210 . . . . . . 7  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  H  Fn  ( S  X.  S
) )
1716ex 432 . . . . . 6  |-  ( ph  ->  ( H  Fn  (
s  X.  s )  ->  H  Fn  ( S  X.  S ) ) )
184, 17syl5 32 . . . . 5  |-  ( ph  ->  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( S  X.  S ) ) )
1918rexlimdvw 2891 . . . 4  |-  ( ph  ->  ( E. s  e. 
~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( S  X.  S
) ) )
2019adantld 465 . . 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 1739 . 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 15 1  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399   E.wex 1627    e. wcel 1836   E.wrex 2747   ~Pcpw 3944   class class class wbr 4384    X. cxp 4928   dom cdm 4930    Fn wfn 5508   ` cfv 5513   X_cixp 7410    C_cat cssc 15236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ixp 7411  df-ssc 15239
This theorem is referenced by:  ssctr  15254  ssceq  15255  subcfn  15270  subsubc  15282
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