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Theorem sscfn1 14019
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 14016 . . 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 190 . 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 7070 . . . . . 6  |-  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( s  X.  s
) )
5 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  H  Fn  ( s  X.  s
) )
6 sscfn1.2 . . . . . . . . . . . 12  |-  ( ph  ->  S  =  dom  dom  H )
76adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  S  =  dom  dom  H )
8 fndm 5546 . . . . . . . . . . . . . 14  |-  ( H  Fn  ( s  X.  s )  ->  dom  H  =  ( s  X.  s ) )
98adantl 454 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  H  =  ( s  X.  s ) )
109dmeqd 5074 . . . . . . . . . . . 12  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  dom  (
s  X.  s ) )
11 dmxpid 5091 . . . . . . . . . . . 12  |-  dom  (
s  X.  s )  =  s
1210, 11syl6eq 2486 . . . . . . . . . . 11  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  s )
137, 12eqtr2d 2471 . . . . . . . . . 10  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  s  =  S )
1413, 13xpeq12d 4905 . . . . . . . . 9  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  (
s  X.  s )  =  ( S  X.  S ) )
1514fneq2d 5539 . . . . . . . 8  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  ( H  Fn  ( s  X.  s )  <->  H  Fn  ( S  X.  S
) ) )
165, 15mpbid 203 . . . . . . 7  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  H  Fn  ( S  X.  S
) )
1716ex 425 . . . . . 6  |-  ( ph  ->  ( H  Fn  (
s  X.  s )  ->  H  Fn  ( S  X.  S ) ) )
184, 17syl5 31 . . . . 5  |-  ( ph  ->  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( S  X.  S ) ) )
1918rexlimdvw 2835 . . . 4  |-  ( ph  ->  ( E. s  e. 
~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x )  ->  H  Fn  ( S  X.  S
) ) )
2019adantld 455 . . 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 1647 . 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 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 4214    X. cxp 4878   dom cdm 4880    Fn wfn 5451   ` cfv 5456   X_cixp 7065    C_cat cssc 14009
This theorem is referenced by:  ssctr  14027  ssceq  14028  subcfn  14040  subsubc  14052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ixp 7066  df-ssc 14012
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