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Theorem ssceq 13752
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 2317 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  A  =  dom  dom  A )
31, 2sscfn1 13743 . . . . 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 2317 . . . . . 6  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  B  =  dom  dom  B )
64, 5sscfn1 13743 . . . . 5  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  B  Fn  ( dom  dom  B  X.  dom  dom  B ) )
73, 6, 1ssc1 13747 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  A  C_ 
dom  dom  B )
86, 3, 4ssc1 13747 . . . 4  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  B  C_ 
dom  dom  A )
97, 8eqssd 3230 . . 3  |-  ( ( A  C_cat  B  /\  B  C_cat  A
)  ->  dom  dom  A  =  dom  dom  B )
109, 9xpeq12d 4751 . 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 13748 . . . 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 3215 . . . . 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 3215 . . . . 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 13748 . . . 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 3230 . . 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 2669 . 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 5992 . . 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 1633    e. wcel 1701   A.wral 2577    C_ wss 3186   class class class wbr 4060    X. cxp 4724   dom cdm 4726    Fn wfn 5287  (class class class)co 5900    C_cat cssc 13733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-ixp 6861  df-ssc 13736
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