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Theorem sscfn2 13981
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 )
sscfn2.2  |-  ( ph  ->  T  =  dom  dom  J )
Assertion
Ref Expression
sscfn2  |-  ( ph  ->  J  Fn  ( T  X.  T ) )

Proof of Theorem sscfn2
Dummy variables  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sscfn1.1 . . 3  |-  ( ph  ->  H  C_cat  J )
2 brssc 13977 . . 3  |-  ( H 
C_cat  J  <->  E. t ( J  Fn  ( t  X.  t )  /\  E. y  e.  ~P  t H  e.  X_ x  e.  ( y  X.  y
) ~P ( J `
 x ) ) )
31, 2sylib 189 . 2  |-  ( ph  ->  E. t ( J  Fn  ( t  X.  t )  /\  E. y  e.  ~P  t H  e.  X_ x  e.  ( y  X.  y
) ~P ( J `
 x ) ) )
4 simpr 448 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( t  X.  t
) )
5 sscfn2.2 . . . . . . . . . 10  |-  ( ph  ->  T  =  dom  dom  J )
65adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  T  =  dom  dom  J )
7 fndm 5511 . . . . . . . . . . . 12  |-  ( J  Fn  ( t  X.  t )  ->  dom  J  =  ( t  X.  t ) )
87adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  J  =  ( t  X.  t ) )
98dmeqd 5039 . . . . . . . . . 10  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  dom  (
t  X.  t ) )
10 dmxpid 5056 . . . . . . . . . 10  |-  dom  (
t  X.  t )  =  t
119, 10syl6eq 2460 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  t )
126, 11eqtr2d 2445 . . . . . . . 8  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  t  =  T )
1312, 12xpeq12d 4870 . . . . . . 7  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  (
t  X.  t )  =  ( T  X.  T ) )
1413fneq2d 5504 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  ( J  Fn  ( t  X.  t )  <->  J  Fn  ( T  X.  T
) ) )
154, 14mpbid 202 . . . . 5  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( T  X.  T
) )
1615ex 424 . . . 4  |-  ( ph  ->  ( J  Fn  (
t  X.  t )  ->  J  Fn  ( T  X.  T ) ) )
1716adantrd 455 . . 3  |-  ( ph  ->  ( ( J  Fn  ( t  X.  t
)  /\  E. y  e.  ~P  t H  e.  X_ x  e.  (
y  X.  y ) ~P ( J `  x ) )  ->  J  Fn  ( T  X.  T ) ) )
1817exlimdv 1643 . 2  |-  ( ph  ->  ( E. t ( J  Fn  ( t  X.  t )  /\  E. y  e.  ~P  t H  e.  X_ x  e.  ( y  X.  y
) ~P ( J `
 x ) )  ->  J  Fn  ( T  X.  T ) ) )
193, 18mpd 15 1  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   E.wrex 2675   ~Pcpw 3767   class class class wbr 4180    X. cxp 4843   dom cdm 4845    Fn wfn 5416   ` cfv 5421   X_cixp 7030    C_cat cssc 13970
This theorem is referenced by:  ssc2  13985  ssctr  13988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ixp 7031  df-ssc 13973
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