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Theorem ssctr 13702
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 443 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  C_cat  B )
2 eqidd 2284 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  =  dom  dom  A )
31, 2sscfn1 13694 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  Fn  ( dom  dom  A  X.  dom  dom  A ) )
4 eqidd 2284 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  =  dom  dom  B )
51, 4sscfn2 13695 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  B  Fn  ( dom  dom  B  X.  dom  dom  B ) )
63, 5, 1ssc1 13698 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  C_ 
dom  dom  B )
7 simpr 447 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  B  C_cat  C )
8 eqidd 2284 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  C  =  dom  dom  C )
97, 8sscfn2 13695 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  Fn  ( dom  dom  C  X.  dom  dom  C ) )
105, 9, 7ssc1 13698 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  C_ 
dom  dom  C )
116, 10sstrd 3189 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  C_ 
dom  dom  C )
123adantr 451 . . . . 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 451 . . . . 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 732 . . . . 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 733 . . . . 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 13699 . . . 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 451 . . . . 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 451 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  B  C_cat  C )
196adantr 451 . . . . . 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 3181 . . . . 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 3181 . . . . 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 13699 . . . 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 3189 . . 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 2635 . 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 13690 . . . . . 6  |-  Rel  C_cat
2625brrelex2i 4730 . . . . 5  |-  ( B 
C_cat  C  ->  C  e.  _V )
2726adantl 452 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  e.  _V )
28 dmexg 4939 . . . 4  |-  ( C  e.  _V  ->  dom  C  e.  _V )
29 dmexg 4939 . . . 4  |-  ( dom 
C  e.  _V  ->  dom 
dom  C  e.  _V )
3027, 28, 293syl 18 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  C  e.  _V )
313, 9, 30isssc 13697 . 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 888 1  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  C_cat  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    X. cxp 4687   dom cdm 4689    Fn wfn 5250  (class class class)co 5858    C_cat cssc 13684
This theorem is referenced by:  subsubc  13727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-ixp 6818  df-ssc 13687
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