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Theorem sscfn1 13710
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 13707 . . 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 6838 . . . . . 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 5359 . . . . . . . . . . . . . 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 4897 . . . . . . . . . . . 12  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  dom  (
s  X.  s ) )
11 dmxpid 4914 . . . . . . . . . . . 12  |-  dom  (
s  X.  s )  =  s
1210, 11syl6eq 2344 . . . . . . . . . . 11  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  dom  dom 
H  =  s )
137, 12eqtr2d 2329 . . . . . . . . . 10  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  s  =  S )
1413, 13xpeq12d 4730 . . . . . . . . 9  |-  ( (
ph  /\  H  Fn  ( s  X.  s
) )  ->  (
s  X.  s )  =  ( S  X.  S ) )
1514fneq2d 5352 . . . . . . . 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 2683 . . . 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 1626 . 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 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   ~Pcpw 3638   class class class wbr 4039    X. cxp 4703   dom cdm 4705    Fn wfn 5266   ` cfv 5271   X_cixp 6833    C_cat cssc 13700
This theorem is referenced by:  ssctr  13718  ssceq  13719  subcfn  13731  subsubc  13743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ixp 6834  df-ssc 13703
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