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Theorem ssctr 13984
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 444 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  C_cat  B )
2 eqidd 2409 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  =  dom  dom  A )
31, 2sscfn1 13976 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  Fn  ( dom  dom  A  X.  dom  dom  A ) )
4 eqidd 2409 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  =  dom  dom  B )
51, 4sscfn2 13977 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  B  Fn  ( dom  dom  B  X.  dom  dom  B ) )
63, 5, 1ssc1 13980 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  C_ 
dom  dom  B )
7 simpr 448 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  B  C_cat  C )
8 eqidd 2409 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  C  =  dom  dom  C )
97, 8sscfn2 13977 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  Fn  ( dom  dom  C  X.  dom  dom  C ) )
105, 9, 7ssc1 13980 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  C_ 
dom  dom  C )
116, 10sstrd 3322 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  C_ 
dom  dom  C )
123adantr 452 . . . . 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 452 . . . . 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 733 . . . . 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 734 . . . . 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 13981 . . . 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 452 . . . . 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 452 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  B  C_cat  C )
196adantr 452 . . . . . 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 3313 . . . . 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 3313 . . . . 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 13981 . . . 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 3322 . . 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 2762 . 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 13972 . . . . . 6  |-  Rel  C_cat
2625brrelex2i 4882 . . . . 5  |-  ( B 
C_cat  C  ->  C  e.  _V )
2726adantl 453 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  e.  _V )
28 dmexg 5093 . . . 4  |-  ( C  e.  _V  ->  dom  C  e.  _V )
29 dmexg 5093 . . . 4  |-  ( dom 
C  e.  _V  ->  dom 
dom  C  e.  _V )
3027, 28, 293syl 19 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  C  e.  _V )
313, 9, 30isssc 13979 . 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 889 1  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  C_cat  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   A.wral 2670   _Vcvv 2920    C_ wss 3284   class class class wbr 4176    X. cxp 4839   dom cdm 4841    Fn wfn 5412  (class class class)co 6044    C_cat cssc 13966
This theorem is referenced by:  subsubc  14009
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-ixp 7027  df-ssc 13969
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