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Theorem ssctr 16406
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 473 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐵)
2 eqidd 2622 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 = dom dom 𝐴)
31, 2sscfn1 16398 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4 eqidd 2622 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 = dom dom 𝐵)
51, 4sscfn2 16399 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
63, 5, 1ssc1 16402 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐵)
7 simpr 477 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐵cat 𝐶)
8 eqidd 2622 . . . . 5 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 = dom dom 𝐶)
97, 8sscfn2 16399 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 Fn (dom dom 𝐶 × dom dom 𝐶))
105, 9, 7ssc1 16402 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐵 ⊆ dom dom 𝐶)
116, 10sstrd 3593 . 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 793 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
15 simprr 795 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
1612, 13, 14, 15ssc2 16403 . . . 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 3584 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
2119, 15sseldd 3584 . . . . 5 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
2217, 18, 20, 21ssc2 16403 . . . 4 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐶𝑦))
2316, 22sstrd 3593 . . 3 (((𝐴cat 𝐵𝐵cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
2423ralrimivva 2965 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
25 sscrel 16394 . . . . . 6 Rel ⊆cat
2625brrelex2i 5119 . . . . 5 (𝐵cat 𝐶𝐶 ∈ V)
2726adantl 482 . . . 4 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐶 ∈ V)
28 dmexg 7044 . . . 4 (𝐶 ∈ V → dom 𝐶 ∈ V)
29 dmexg 7044 . . . 4 (dom 𝐶 ∈ V → dom dom 𝐶 ∈ V)
3027, 28, 293syl 18 . . 3 ((𝐴cat 𝐵𝐵cat 𝐶) → dom dom 𝐶 ∈ V)
313, 9, 30isssc 16401 . 2 ((𝐴cat 𝐵𝐵cat 𝐶) → (𝐴cat 𝐶 ↔ (dom dom 𝐴 ⊆ dom dom 𝐶 ∧ ∀𝑥 ∈ dom dom 𝐴𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))))
3211, 24, 31mpbir2and 956 1 ((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907  Vcvv 3186  wss 3555   class class class wbr 4613   × cxp 5072  dom cdm 5074   Fn wfn 5842  (class class class)co 6604  cat cssc 16388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-ixp 7853  df-ssc 16391
This theorem is referenced by:  subsubc  16434
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