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Theorem sscfn2 15243
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 15239 . . 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 196 . 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 459 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( t  X.  t
) )
5 sscfn2.2 . . . . . . . . . 10  |-  ( ph  ->  T  =  dom  dom  J )
65adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  T  =  dom  dom  J )
7 fndm 5601 . . . . . . . . . . . 12  |-  ( J  Fn  ( t  X.  t )  ->  dom  J  =  ( t  X.  t ) )
87adantl 464 . . . . . . . . . . 11  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  J  =  ( t  X.  t ) )
98dmeqd 5131 . . . . . . . . . 10  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  dom  (
t  X.  t ) )
10 dmxpid 5148 . . . . . . . . . 10  |-  dom  (
t  X.  t )  =  t
119, 10syl6eq 2449 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  t )
126, 11eqtr2d 2434 . . . . . . . 8  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  t  =  T )
1312sqxpeqd 4952 . . . . . . 7  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  (
t  X.  t )  =  ( T  X.  T ) )
1413fneq2d 5593 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  ( J  Fn  ( t  X.  t )  <->  J  Fn  ( T  X.  T
) ) )
154, 14mpbid 210 . . . . 5  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( T  X.  T
) )
1615ex 432 . . . 4  |-  ( ph  ->  ( J  Fn  (
t  X.  t )  ->  J  Fn  ( T  X.  T ) ) )
1716adantrd 466 . . 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 1739 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399   E.wex 1627    e. wcel 1836   E.wrex 2743   ~Pcpw 3940   class class class wbr 4380    X. cxp 4924   dom cdm 4926    Fn wfn 5504   ` cfv 5509   X_cixp 7406    C_cat cssc 15232
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 2016  ax-ext 2370  ax-rep 4491  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
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 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-reu 2749  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-ixp 7407  df-ssc 15235
This theorem is referenced by:  ssc2  15247  ssctr  15250
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