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Theorem ssctr 17083
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 483 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐵)
2 eqidd 2819 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 17075 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 eqidd 2819 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 = dom dom 𝐵)
51, 4sscfn2 17076 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
63, 5, 1ssc1 17079 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐵)
7 simpr 485 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵cat 𝐶)
8 eqidd 2819 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 = dom dom 𝐶)
97, 8sscfn2 17076 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 Fn (dom dom 𝐶 × dom dom 𝐶))
105, 9, 7ssc1 17079 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 ⊆ dom dom 𝐶)
116, 10sstrd 3974 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐶)
123adantr 481 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
131adantr 481 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴cat 𝐵)
14 simprl 767 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
15 simprr 769 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
1612, 13, 14, 15ssc2 17080 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
175adantr 481 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
187adantr 481 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵cat 𝐶)
196adantr 481 . . . . . 6 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
2019, 14sseldd 3965 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2119, 15sseldd 3965 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2217, 18, 20, 21ssc2 17080 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐶𝑦))
2316, 22sstrd 3974 . . 3 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
2423ralrimivva 3188 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
25 sscrel 17071 . . . . . 6 Rel ⊆cat
2625brrelex2i 5602 . . . . 5 (𝐵cat 𝐶𝐶 ∈ V)
2726adantl 482 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 ∈ V)
28 dmexg 7602 . . . 4 (𝐶 ∈ V → dom 𝐶 ∈ V)
29 dmexg 7602 . . . 4 (dom 𝐶 ∈ V → dom dom 𝐶 ∈ V)
3027, 28, 293syl 18 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 ∈ V)
313, 9, 30isssc 17078 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → (𝐴cat 𝐶 ↔ (dom dom 𝐴 ⊆ dom dom 𝐶 ∧ ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))))
3211, 24, 31mpbir2and 709 1 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wral 3135  Vcvv 3492  wss 3933   class class class wbr 5057   × cxp 5546  dom cdm 5548   Fn wfn 6343  (class class class)co 7145  cat cssc 17065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-ixp 8450  df-ssc 17068
This theorem is referenced by:  subsubc  17111
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