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Theorem ssceq 14026
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 444 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  A  C_cat  B )
2 eqidd 2437 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  A  =  dom  dom  A )
31, 2sscfn1 14017 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  A  Fn  ( dom  dom  A  X.  dom  dom  A ) )
4 simpr 448 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  B  C_cat  A )
5 eqidd 2437 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  B  =  dom  dom  B )
64, 5sscfn1 14017 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  B  Fn  ( dom  dom  B  X.  dom  dom  B ) )
73, 6, 1ssc1 14021 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  A  C_ 
dom  dom  B )
86, 3, 4ssc1 14021 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  B  C_ 
dom  dom  A )
97, 8eqssd 3365 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  A  =  dom  dom  B )
109, 9xpeq12d 4903 . 2  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  ( dom  dom 
A  X.  dom  dom  A )  =  ( dom 
dom  B  X.  dom  dom  B ) )
113adantr 452 . . . . 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 452 . . . . 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 733 . . . . 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 734 . . . . 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 14022 . . . 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 452 . . . . 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 452 . . . . 5  |-  ( ( ( A  C_cat  B  /\  B  C_cat  A )  /\  (
x  e.  dom  dom  A  /\  y  e.  dom  dom 
A ) )  ->  B  C_cat  A )
187adantr 452 . . . . . 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 3349 . . . . 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 3349 . . . . 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 14022 . . . 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 3365 . . 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 2798 . 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 6176 . . 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 643 . 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 889 1  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   class class class wbr 4212    X. cxp 4876   dom cdm 4878    Fn wfn 5449  (class class class)co 6081    C_cat cssc 14007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-ixp 7064  df-ssc 14010
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