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Theorem ssceq 17526
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 483 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴cat 𝐵)
2 eqidd 2739 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 17517 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 simpr 485 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵cat 𝐴)
5 eqidd 2739 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 = dom dom 𝐵)
64, 5sscfn1 17517 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
73, 6, 1ssc1 17521 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 ⊆ dom dom 𝐵)
86, 3, 4ssc1 17521 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 ⊆ dom dom 𝐴)
97, 8eqssd 3938 . . 3 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐵)
109sqxpeqd 5617 . 2 ((𝐴cat 𝐵𝐵cat 𝐴) → (dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵))
113adantr 481 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
121adantr 481 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴cat 𝐵)
13 simprl 768 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
14 simprr 770 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
1511, 12, 13, 14ssc2 17522 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
166adantr 481 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
174adantr 481 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵cat 𝐴)
187adantr 481 . . . . . 6 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
1918, 13sseldd 3922 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2018, 14sseldd 3922 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2116, 17, 19, 20ssc2 17522 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐴𝑦))
2215, 21eqssd 3938 . . 3 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) = (𝑥𝐵𝑦))
2322ralrimivva 3120 . 2 ((𝐴cat 𝐵𝐵cat 𝐴) → ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) = (𝑥𝐵𝑦))
24 eqfnov 7394 . . 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 584 . 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 710 1 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  wss 3887   class class class wbr 5074   × cxp 5583  dom cdm 5585   Fn wfn 6422  (class class class)co 7268  cat cssc 17507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-ov 7271  df-ixp 8674  df-ssc 17510
This theorem is referenced by: (None)
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