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Theorem ssctr 13795
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 2359 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  A  =  dom  dom  A )
31, 2sscfn1 13787 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  A  Fn  ( dom  dom  A  X.  dom  dom  A ) )
4 eqidd 2359 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  =  dom  dom  B )
51, 4sscfn2 13788 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  B  Fn  ( dom  dom  B  X.  dom  dom  B ) )
63, 5, 1ssc1 13791 . . 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 2359 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  C  =  dom  dom  C )
97, 8sscfn2 13788 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  Fn  ( dom  dom  C  X.  dom  dom  C ) )
105, 9, 7ssc1 13791 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  dom  dom  B  C_ 
dom  dom  C )
116, 10sstrd 3265 . 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 13792 . . . 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 3257 . . . . 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 3257 . . . . 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 13792 . . . 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 3265 . . 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 2711 . 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 13783 . . . . . 6  |-  Rel  C_cat
2625brrelex2i 4809 . . . . 5  |-  ( B 
C_cat  C  ->  C  e.  _V )
2726adantl 452 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  C
)  ->  C  e.  _V )
28 dmexg 5018 . . . 4  |-  ( C  e.  _V  ->  dom  C  e.  _V )
29 dmexg 5018 . . . 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 13790 . 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 1710   A.wral 2619   _Vcvv 2864    C_ wss 3228   class class class wbr 4102    X. cxp 4766   dom cdm 4768    Fn wfn 5329  (class class class)co 5942    C_cat cssc 13777
This theorem is referenced by:  subsubc  13820
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-ov 5945  df-ixp 6903  df-ssc 13780
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