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Theorem ssctr 17091
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 486 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐵)
2 eqidd 2825 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 17083 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 eqidd 2825 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 = dom dom 𝐵)
51, 4sscfn2 17084 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
63, 5, 1ssc1 17087 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐵)
7 simpr 488 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵cat 𝐶)
8 eqidd 2825 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 = dom dom 𝐶)
97, 8sscfn2 17084 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 Fn (dom dom 𝐶 × dom dom 𝐶))
105, 9, 7ssc1 17087 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 ⊆ dom dom 𝐶)
116, 10sstrd 3962 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐶)
123adantr 484 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
131adantr 484 . . . . 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 17088 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
175adantr 484 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
187adantr 484 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵cat 𝐶)
196adantr 484 . . . . . 6 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
2019, 14sseldd 3953 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2119, 15sseldd 3953 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2217, 18, 20, 21ssc2 17088 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐶𝑦))
2316, 22sstrd 3962 . . 3 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
2423ralrimivva 3186 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
25 sscrel 17079 . . . . . 6 Rel ⊆cat
2625brrelex2i 5596 . . . . 5 (𝐵cat 𝐶𝐶 ∈ V)
2726adantl 485 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 ∈ V)
28 dmexg 7603 . . . 4 (𝐶 ∈ V → dom 𝐶 ∈ V)
29 dmexg 7603 . . . 4 (dom 𝐶 ∈ V → dom dom 𝐶 ∈ V)
3027, 28, 293syl 18 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 ∈ V)
313, 9, 30isssc 17086 . 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 399  wcel 2115  wral 3133  Vcvv 3480  wss 3919   class class class wbr 5052   × cxp 5540  dom cdm 5542   Fn wfn 6338  (class class class)co 7145  cat cssc 17073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7148  df-ixp 8452  df-ssc 17076
This theorem is referenced by:  subsubc  17119
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