Theorem ssctr 17787

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 2730	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐴 = dom dom 𝐴)
3	1, 2	sscfn1 17779	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4		eqidd 2730	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐵 = dom dom 𝐵)
5	1, 4	sscfn2 17780	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
6	3, 5, 1	ssc1 17783	. . 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 2730	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐶 = dom dom 𝐶)
9	7, 8	sscfn2 17780	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐶 Fn (dom dom 𝐶 × dom dom 𝐶))
10	5, 9, 7	ssc1 17783	. . 3 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐵 ⊆ dom dom 𝐶)
11	6, 10	sstrd 3957	. 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 770	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
15		simprr 772	. . . . 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 17784	. . . 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 3947	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
21	19, 15	sseldd 3947	. . . . 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 17784	. . . 4 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐶𝑦))
23	16, 22	sstrd 3957	. . 3 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
24	23	ralrimivva 3180	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → ∀𝑥 ∈ dom dom 𝐴∀𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
25		sscrel 17775	. . . . . 6 ⊢ Rel ⊆cat
26	25	brrelex2i 5695	. . . . 5 ⊢ (𝐵 ⊆cat 𝐶 → 𝐶 ∈ V)
27	26	adantl 481	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐶 ∈ V)
28		dmexg 7877	. . . 4 ⊢ (𝐶 ∈ V → dom 𝐶 ∈ V)
29		dmexg 7877	. . . 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 17782	. 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 2109  ∀wral 3044  Vcvv 3447   ⊆ wss 3914   class class class wbr 5107   × cxp 5636  dom cdm 5638   Fn wfn 6506  (class class class)co 7387   ⊆_cat cssc 17769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-ixp 8871  df-ssc 17772

This theorem is referenced by:  subsubc  17815