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Theorem ssctr 15730
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 459 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  C_cat  B )
2 eqidd 2452 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  =  dom  dom  A )
31, 2sscfn1 15722 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  Fn  ( dom  dom  A  X.  dom  dom  A ) )
4 eqidd 2452 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  =  dom  dom  B )
51, 4sscfn2 15723 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  B  Fn  ( dom  dom  B  X.  dom  dom  B ) )
63, 5, 1ssc1 15726 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  C_ 
dom  dom  B )
7 simpr 463 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  B  C_cat  C )
8 eqidd 2452 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  C  =  dom  dom  C )
97, 8sscfn2 15723 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  Fn  ( dom  dom  C  X.  dom  dom  C ) )
105, 9, 7ssc1 15726 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  C_ 
dom  dom  C )
116, 10sstrd 3442 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  C_ 
dom  dom  C )
123adantr 467 . . . . 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 467 . . . . 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 764 . . . . 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 766 . . . . 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 15727 . . . 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 467 . . . . 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 467 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  C )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  B  C_cat  C )
196adantr 467 . . . . . 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 3433 . . . . 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 3433 . . . . 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 15727 . . . 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 3442 . . 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 2809 . 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 15718 . . . . . 6  |-  Rel  C_cat
2625brrelex2i 4876 . . . . 5  |-  ( B 
C_cat  C  ->  C  e.  _V )
2726adantl 468 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  e.  _V )
28 dmexg 6724 . . . 4  |-  ( C  e.  _V  ->  dom  C  e.  _V )
29 dmexg 6724 . . . 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 15725 . 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 933 1  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  C_cat  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1887   A.wral 2737   _Vcvv 3045    C_ wss 3404   class class class wbr 4402    X. cxp 4832   dom cdm 4834    Fn wfn 5577  (class class class)co 6290    C_cat cssc 15712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-ixp 7523  df-ssc 15715
This theorem is referenced by:  subsubc  15758
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