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Theorem ssceq 13703
Description: The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
ssceq  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  A  =  B )

Proof of Theorem ssceq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  A  C_cat  B )
2 eqidd 2284 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  A  =  dom  dom  A )
31, 2sscfn1 13694 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  A  Fn  ( dom  dom  A  X.  dom  dom  A ) )
4 simpr 447 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  B  C_cat  A )
5 eqidd 2284 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  B  =  dom  dom  B )
64, 5sscfn1 13694 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  B  Fn  ( dom  dom  B  X.  dom  dom  B ) )
73, 6, 1ssc1 13698 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  A  C_ 
dom  dom  B )
86, 3, 4ssc1 13698 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  B  C_ 
dom  dom  A )
97, 8eqssd 3196 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  A  =  dom  dom  B )
109, 9xpeq12d 4714 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  ( dom  dom 
A  X.  dom  dom  A )  =  ( dom 
dom  B  X.  dom  dom  B ) )
113adantr 451 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  A  Fn  ( dom  dom 
A  X.  dom  dom  A ) )
121adantr 451 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  A  C_cat  B )
13 simprl 732 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  x  e.  dom  dom  A
)
14 simprr 733 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
y  e.  dom  dom  A )
1511, 12, 13, 14ssc2 13699 . . . 4  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
( x A y )  C_  ( x B y ) )
166adantr 451 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  B  Fn  ( dom  dom 
B  X.  dom  dom  B ) )
174adantr 451 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  B  C_cat  A )
187adantr 451 . . . . . 6  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  dom  dom  A  C_  dom  dom 
B )
1918, 13sseldd 3181 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  x  e.  dom  dom  B
)
2018, 14sseldd 3181 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
y  e.  dom  dom  B )
2116, 17, 19, 20ssc2 13699 . . . 4  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
( x B y )  C_  ( x A y ) )
2215, 21eqssd 3196 . . 3  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  -> 
( x A y )  =  ( x B y ) )
2322ralrimivva 2635 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  A. x  e.  dom  dom  A A. y  e.  dom  dom  A
( x A y )  =  ( x B y ) )
24 eqfnov 5950 . . 3  |-  ( ( A  Fn  ( dom 
dom  A  X.  dom  dom  A )  /\  B  Fn  ( dom  dom  B  X.  dom  dom  B ) )  ->  ( A  =  B  <->  ( ( dom 
dom  A  X.  dom  dom  A )  =  ( dom 
dom  B  X.  dom  dom  B )  /\  A. x  e.  dom  dom  A A. y  e.  dom  dom  A
( x A y )  =  ( x B y ) ) ) )
253, 6, 24syl2anc 642 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  ( A  =  B  <->  ( ( dom 
dom  A  X.  dom  dom  A )  =  ( dom 
dom  B  X.  dom  dom  B )  /\  A. x  e.  dom  dom  A A. y  e.  dom  dom  A
( x A y )  =  ( x B y ) ) ) )
2610, 23, 25mpbir2and 888 1  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    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 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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