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Theorem sscfn1 14850
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 14847 . . 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 7380 . . . . . 6  |-  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( s  X.  s
) )
5 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  H  Fn  ( s  X.  s
) )
6 sscfn1.2 . . . . . . . . . . . 12  |-  ( ph  ->  S  =  dom  dom  H )
76adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  S  =  dom  dom  H )
8 fndm 5619 . . . . . . . . . . . . . 14  |-  ( H  Fn  ( s  X.  s )  ->  dom  H  =  ( s  X.  s ) )
98adantl 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  H  =  ( s  X.  s ) )
109dmeqd 5151 . . . . . . . . . . . 12  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  dom  (
s  X.  s ) )
11 dmxpid 5168 . . . . . . . . . . . 12  |-  dom  (
s  X.  s )  =  s
1210, 11syl6eq 2511 . . . . . . . . . . 11  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  s )
137, 12eqtr2d 2496 . . . . . . . . . 10  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  s  =  S )
1413, 13xpeq12d 4974 . . . . . . . . 9  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  (
s  X.  s )  =  ( S  X.  S ) )
1514fneq2d 5611 . . . . . . . 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 434 . . . . . 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 2950 . . . 4  |-  ( ph  ->  ( E. s  e. 
~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( S  X.  S
) ) )
2019adantld 467 . . 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 1691 . 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 369    = wceq 1370   E.wex 1587    e. wcel 1758   E.wrex 2800   ~Pcpw 3969   class class class wbr 4401    X. cxp 4947   dom cdm 4949    Fn wfn 5522   ` cfv 5527   X_cixp 7374    C_cat cssc 14840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ixp 7375  df-ssc 14843
This theorem is referenced by:  ssctr  14858  ssceq  14859  subcfn  14871  subsubc  14883
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