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Theorem ssceq 16693
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 468 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴cat 𝐵)
2 eqidd 2772 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 16684 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 simpr 471 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵cat 𝐴)
5 eqidd 2772 . . . . . 6 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 = dom dom 𝐵)
64, 5sscfn1 16684 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
73, 6, 1ssc1 16688 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 ⊆ dom dom 𝐵)
86, 3, 4ssc1 16688 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐵 ⊆ dom dom 𝐴)
97, 8eqssd 3769 . . 3 ((𝐴cat 𝐵𝐵cat 𝐴) → dom dom 𝐴 = dom dom 𝐵)
109sqxpeqd 5281 . 2 ((𝐴cat 𝐵𝐵cat 𝐴) → (dom dom 𝐴 × dom dom 𝐴) = (dom dom 𝐵 × dom dom 𝐵))
113adantr 466 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
121adantr 466 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴cat 𝐵)
13 simprl 754 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
14 simprr 756 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
1511, 12, 13, 14ssc2 16689 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
166adantr 466 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
174adantr 466 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵cat 𝐴)
187adantr 466 . . . . . 6 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
1918, 13sseldd 3753 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2018, 14sseldd 3753 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2116, 17, 19, 20ssc2 16689 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐴) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐴𝑦))
2215, 21eqssd 3769 . . 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 6917 . . 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 573 . 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 692 1 ((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wss 3723   class class class wbr 4787   × cxp 5248  dom cdm 5250   Fn wfn 6025  (class class class)co 6796  cat cssc 16674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-ixp 8067  df-ssc 16677
This theorem is referenced by: (None)
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