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Theorem ssctr 17799
Description: The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
ssctr ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐶)

Proof of Theorem ssctr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐵)
2 eqidd 2728 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 17791 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 eqidd 2728 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 = dom dom 𝐵)
51, 4sscfn2 17792 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
63, 5, 1ssc1 17795 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐵)
7 simpr 484 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵cat 𝐶)
8 eqidd 2728 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 = dom dom 𝐶)
97, 8sscfn2 17792 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 Fn (dom dom 𝐶 × dom dom 𝐶))
105, 9, 7ssc1 17795 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 ⊆ dom dom 𝐶)
116, 10sstrd 3988 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐶)
123adantr 480 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
131adantr 480 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴cat 𝐵)
14 simprl 770 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
15 simprr 772 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
1612, 13, 14, 15ssc2 17796 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
175adantr 480 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
187adantr 480 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵cat 𝐶)
196adantr 480 . . . . . 6 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
2019, 14sseldd 3979 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2119, 15sseldd 3979 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2217, 18, 20, 21ssc2 17796 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐶𝑦))
2316, 22sstrd 3988 . . 3 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
2423ralrimivva 3195 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
25 sscrel 17787 . . . . . 6 Rel ⊆cat
2625brrelex2i 5729 . . . . 5 (𝐵cat 𝐶𝐶 ∈ V)
2726adantl 481 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 ∈ V)
28 dmexg 7903 . . . 4 (𝐶 ∈ V → dom 𝐶 ∈ V)
29 dmexg 7903 . . . 4 (dom 𝐶 ∈ V → dom dom 𝐶 ∈ V)
3027, 28, 293syl 18 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 ∈ V)
313, 9, 30isssc 17794 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → (𝐴cat 𝐶 ↔ (dom dom 𝐴 ⊆ dom dom 𝐶 ∧ ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))))
3211, 24, 31mpbir2and 712 1 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  wral 3056  Vcvv 3469  wss 3944   class class class wbr 5142   × cxp 5670  dom cdm 5672   Fn wfn 6537  (class class class)co 7414  cat cssc 17781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-ixp 8908  df-ssc 17784
This theorem is referenced by:  subsubc  17830
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