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Theorem ssctr 16691
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 468 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐵)
2 eqidd 2771 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 16683 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 eqidd 2771 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 = dom dom 𝐵)
51, 4sscfn2 16684 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
63, 5, 1ssc1 16687 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐵)
7 simpr 471 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵cat 𝐶)
8 eqidd 2771 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 = dom dom 𝐶)
97, 8sscfn2 16684 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 Fn (dom dom 𝐶 × dom dom 𝐶))
105, 9, 7ssc1 16687 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 ⊆ dom dom 𝐶)
116, 10sstrd 3760 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐶)
123adantr 466 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
131adantr 466 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐴cat 𝐵)
14 simprl 746 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
15 simprr 748 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
1612, 13, 14, 15ssc2 16688 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
175adantr 466 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
187adantr 466 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝐵cat 𝐶)
196adantr 466 . . . . . 6 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
2019, 14sseldd 3751 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2119, 15sseldd 3751 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2217, 18, 20, 21ssc2 16688 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐶𝑦))
2316, 22sstrd 3760 . . 3 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
2423ralrimivva 3119 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
25 sscrel 16679 . . . . . 6 Rel ⊆cat
2625brrelex2i 5299 . . . . 5 (𝐵cat 𝐶𝐶 ∈ V)
2726adantl 467 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 ∈ V)
28 dmexg 7243 . . . 4 (𝐶 ∈ V → dom 𝐶 ∈ V)
29 dmexg 7243 . . . 4 (dom 𝐶 ∈ V → dom dom 𝐶 ∈ V)
3027, 28, 293syl 18 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 ∈ V)
313, 9, 30isssc 16686 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → (𝐴cat 𝐶 ↔ (dom dom 𝐴 ⊆ dom dom 𝐶 ∧ ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))))
3211, 24, 31mpbir2and 684 1 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2144  wral 3060  Vcvv 3349  wss 3721   class class class wbr 4784   × cxp 5247  dom cdm 5249   Fn wfn 6026  (class class class)co 6792  cat cssc 16673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-ixp 8062  df-ssc 16676
This theorem is referenced by:  subsubc  16719
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