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Theorem ssceq 16533
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 2652 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 16524 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 simpr 476 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵cat 𝐴)
5 eqidd 2652 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 = dom dom 𝐵)
64, 5sscfn1 16524 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
73, 6, 1ssc1 16528 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 ⊆ dom dom 𝐵)
86, 3, 4ssc1 16528 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 ⊆ dom dom 𝐴)
97, 8eqssd 3653 . . 3 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐵)
109sqxpeqd 5175 . 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 809 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
14 simprr 811 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
1511, 12, 13, 14ssc2 16529 . . . 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 3637 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2018, 14sseldd 3637 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2116, 17, 19, 20ssc2 16529 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐴𝑦))
2215, 21eqssd 3653 . . 3 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) = (𝑥𝐵𝑦))
2322ralrimivva 3000 . 2 ((𝐴cat 𝐵𝐵cat 𝐴) → ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))
24 eqfnov 6808 . . 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 694 . 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 977 1 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wss 3607   class class class wbr 4685   × cxp 5141  dom cdm 5143   Fn wfn 5921  (class class class)co 6690  cat cssc 16514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-ixp 7951  df-ssc 16517
This theorem is referenced by: (None)
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