Theorem ssctr 17873

Description: The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
ssctr	⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐴 ⊆cat 𝐶)

Proof of Theorem ssctr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐴 ⊆cat 𝐵)
2		eqidd 2736	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐴 = dom dom 𝐴)
3	1, 2	sscfn1 17865	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4		eqidd 2736	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐵 = dom dom 𝐵)
5	1, 4	sscfn2 17866	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
6	3, 5, 1	ssc1 17869	. . 3 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐵)
7		simpr 484	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐵 ⊆cat 𝐶)
8		eqidd 2736	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐶 = dom dom 𝐶)
9	7, 8	sscfn2 17866	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐶 Fn (dom dom 𝐶 × dom dom 𝐶))
10	5, 9, 7	ssc1 17869	. . 3 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐵 ⊆ dom dom 𝐶)
11	6, 10	sstrd 4006	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐶)
12	3	adantr 480	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
13	1	adantr 480	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐴 ⊆cat 𝐵)
14		simprl 771	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
15		simprr 773	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐴)
16	12, 13, 14, 15	ssc2 17870	. . . 4 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
17	5	adantr 480	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
18	7	adantr 480	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐵 ⊆cat 𝐶)
19	6	adantr 480	. . . . . 6 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → dom dom 𝐴 ⊆ dom dom 𝐵)
20	19, 14	sseldd 3996	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
21	19, 15	sseldd 3996	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑦 ∈ dom dom 𝐵)
22	17, 18, 20, 21	ssc2 17870	. . . 4 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐶𝑦))
23	16, 22	sstrd 4006	. . 3 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
24	23	ralrimivva 3200	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → ∀𝑥 ∈ dom dom 𝐴∀𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
25		sscrel 17861	. . . . . 6 ⊢ Rel ⊆cat
26	25	brrelex2i 5746	. . . . 5 ⊢ (𝐵 ⊆cat 𝐶 → 𝐶 ∈ V)
27	26	adantl 481	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐶 ∈ V)
28		dmexg 7924	. . . 4 ⊢ (𝐶 ∈ V → dom 𝐶 ∈ V)
29		dmexg 7924	. . . 4 ⊢ (dom 𝐶 ∈ V → dom dom 𝐶 ∈ V)
30	27, 28, 29	3syl 18	. . 3 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐶 ∈ V)
31	3, 9, 30	isssc 17868	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → (𝐴 ⊆cat 𝐶 ↔ (dom dom 𝐴 ⊆ dom dom 𝐶 ∧ ∀𝑥 ∈ dom dom 𝐴∀𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))))
32	11, 24, 31	mpbir2and 713	1 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐴 ⊆cat 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ⊆ wss 3963   class class class wbr 5148   × cxp 5687  dom cdm 5689   Fn wfn 6558  (class class class)co 7431   ⊆_cat cssc 17855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-ixp 8937  df-ssc 17858

This theorem is referenced by:  subsubc  17904