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Theorem ssceq 16258
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 472 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴cat 𝐵)
2 eqidd 2611 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 16249 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 simpr 476 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵cat 𝐴)
5 eqidd 2611 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 = dom dom 𝐵)
64, 5sscfn1 16249 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
73, 6, 1ssc1 16253 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 ⊆ dom dom 𝐵)
86, 3, 4ssc1 16253 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 ⊆ dom dom 𝐴)
97, 8eqssd 3585 . . 3 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐵)
109sqxpeqd 5055 . 2 ((𝐴cat 𝐵𝐵cat 𝐴) → (dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵))
113adantr 480 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
121adantr 480 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴cat 𝐵)
13 simprl 790 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
14 simprr 792 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
1511, 12, 13, 14ssc2 16254 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
166adantr 480 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
174adantr 480 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵cat 𝐴)
187adantr 480 . . . . . 6 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
1918, 13sseldd 3569 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2018, 14sseldd 3569 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2116, 17, 19, 20ssc2 16254 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐴𝑦))
2215, 21eqssd 3585 . . 3 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) = (𝑥𝐵𝑦))
2322ralrimivva 2954 . 2 ((𝐴cat 𝐵𝐵cat 𝐴) → ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))
24 eqfnov 6642 . . 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 691 . 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 959 1 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wss 3540   class class class wbr 4578   × cxp 5026  dom cdm 5028   Fn wfn 5785  (class class class)co 6527  cat cssc 16239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-ixp 7773  df-ssc 16242
This theorem is referenced by: (None)
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