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Theorem sscfn2 13788
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 13784 . . 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 188 . 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 447 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( t  X.  t
) )
5 sscfn2.2 . . . . . . . . . 10  |-  ( ph  ->  T  =  dom  dom  J )
65adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  T  =  dom  dom  J )
7 fndm 5422 . . . . . . . . . . . 12  |-  ( J  Fn  ( t  X.  t )  ->  dom  J  =  ( t  X.  t ) )
84, 7syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  J  =  ( t  X.  t ) )
98dmeqd 4960 . . . . . . . . . 10  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  dom  (
t  X.  t ) )
10 dmxpid 4977 . . . . . . . . . 10  |-  dom  (
t  X.  t )  =  t
119, 10syl6eq 2406 . . . . . . . . 9  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  dom  dom 
J  =  t )
126, 11eqtr2d 2391 . . . . . . . 8  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  t  =  T )
1312, 12xpeq12d 4793 . . . . . . 7  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  (
t  X.  t )  =  ( T  X.  T ) )
1413fneq2d 5415 . . . . . 6  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  ( J  Fn  ( t  X.  t )  <->  J  Fn  ( T  X.  T
) ) )
154, 14mpbid 201 . . . . 5  |-  ( (
ph  /\  J  Fn  ( t  X.  t
) )  ->  J  Fn  ( T  X.  T
) )
1615ex 423 . . . 4  |-  ( ph  ->  ( J  Fn  (
t  X.  t )  ->  J  Fn  ( T  X.  T ) ) )
1716adantrd 454 . . 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 1636 . 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 14 1  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   E.wrex 2620   ~Pcpw 3701   class class class wbr 4102    X. cxp 4766   dom cdm 4768    Fn wfn 5329   ` cfv 5334   X_cixp 6902    C_cat cssc 13777
This theorem is referenced by:  ssc2  13792  ssctr  13795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ixp 6903  df-ssc 13780
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