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Theorem ssctr 15241
Description: The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
ssctr  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  C_cat  C )

Proof of Theorem ssctr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  C_cat  B )
2 eqidd 2458 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  =  dom  dom  A )
31, 2sscfn1 15233 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  Fn  ( dom  dom  A  X.  dom  dom  A ) )
4 eqidd 2458 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  =  dom  dom  B )
51, 4sscfn2 15234 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  B  Fn  ( dom  dom  B  X.  dom  dom  B ) )
63, 5, 1ssc1 15237 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  C_ 
dom  dom  B )
7 simpr 461 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  B  C_cat  C )
8 eqidd 2458 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  C  =  dom  dom  C )
97, 8sscfn2 15234 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  Fn  ( dom  dom  C  X.  dom  dom  C ) )
105, 9, 7ssc1 15237 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  C_ 
dom  dom  C )
116, 10sstrd 3509 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  C_ 
dom  dom  C )
123adantr 465 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  A  Fn  ( dom  dom 
A  X.  dom  dom  A ) )
131adantr 465 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  A  C_cat  B )
14 simprl 756 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  x  e.  dom  dom  A
)
15 simprr 757 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
y  e.  dom  dom  A )
1612, 13, 14, 15ssc2 15238 . . . 4  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
( x A y )  C_  ( x B y ) )
175adantr 465 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  B  Fn  ( dom  dom 
B  X.  dom  dom  B ) )
187adantr 465 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  B  C_cat  C )
196adantr 465 . . . . . 6  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  dom  dom  A  C_  dom  dom 
B )
2019, 14sseldd 3500 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  x  e.  dom  dom  B
)
2119, 15sseldd 3500 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
y  e.  dom  dom  B )
2217, 18, 20, 21ssc2 15238 . . . 4  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
( x B y )  C_  ( x C y ) )
2316, 22sstrd 3509 . . 3  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
( x A y )  C_  ( x C y ) )
2423ralrimivva 2878 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A. x  e.  dom  dom  A A. y  e.  dom  dom  A
( x A y )  C_  ( x C y ) )
25 sscrel 15229 . . . . . 6  |-  Rel  C_cat
2625brrelex2i 5050 . . . . 5  |-  ( B 
C_cat  C  ->  C  e.  _V )
2726adantl 466 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  e.  _V )
28 dmexg 6730 . . . 4  |-  ( C  e.  _V  ->  dom  C  e.  _V )
29 dmexg 6730 . . . 4  |-  ( dom 
C  e.  _V  ->  dom 
dom  C  e.  _V )
3027, 28, 293syl 20 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  C  e.  _V )
313, 9, 30isssc 15236 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  ( A  C_cat  C  <-> 
( dom  dom  A  C_  dom  dom  C  /\  A. x  e.  dom  dom  A A. y  e.  dom  dom 
A ( x A y )  C_  (
x C y ) ) ) )
3211, 24, 31mpbir2and 922 1  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  C_cat  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   A.wral 2807   _Vcvv 3109    C_ wss 3471   class class class wbr 4456    X. cxp 5006   dom cdm 5008    Fn wfn 5589  (class class class)co 6296    C_cat cssc 15223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-ixp 7489  df-ssc 15226
This theorem is referenced by:  subsubc  15269
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