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Theorem ssceq 16967
Description: The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
ssceq ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)

Proof of Theorem ssceq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 475 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴cat 𝐵)
2 eqidd 2774 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 16958 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 simpr 477 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵cat 𝐴)
5 eqidd 2774 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 = dom dom 𝐵)
64, 5sscfn1 16958 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
73, 6, 1ssc1 16962 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 ⊆ dom dom 𝐵)
86, 3, 4ssc1 16962 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 ⊆ dom dom 𝐴)
97, 8eqssd 3870 . . 3 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐵)
109sqxpeqd 5436 . 2 ((𝐴cat 𝐵𝐵cat 𝐴) → (dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵))
113adantr 473 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
121adantr 473 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴cat 𝐵)
13 simprl 759 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
14 simprr 761 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
1511, 12, 13, 14ssc2 16963 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
166adantr 473 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
174adantr 473 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵cat 𝐴)
187adantr 473 . . . . . 6 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
1918, 13sseldd 3854 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2018, 14sseldd 3854 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2116, 17, 19, 20ssc2 16963 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐴𝑦))
2215, 21eqssd 3870 . . 3 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) = (𝑥𝐵𝑦))
2322ralrimivva 3136 . 2 ((𝐴cat 𝐵𝐵cat 𝐴) → ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))
24 eqfnov 7095 . . 3 ((𝐴 Fn (dom dom 𝐴 × dom dom 𝐴) ∧ 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵)) → (𝐴 = 𝐵 ↔ ((dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵) ∧ ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))))
253, 6, 24syl2anc 576 . 2 ((𝐴cat 𝐵𝐵cat 𝐴) → (𝐴 = 𝐵 ↔ ((dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵) ∧ ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))))
2610, 23, 25mpbir2and 701 1 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  wral 3083  wss 3824   class class class wbr 4926   × cxp 5402  dom cdm 5404   Fn wfn 6181  (class class class)co 6975  cat cssc 16948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-ov 6978  df-ixp 8259  df-ssc 16951
This theorem is referenced by: (None)
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