Theorem ssctr 17097

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 485	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐴 ⊆cat 𝐵)
2		eqidd 2824	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐴 = dom dom 𝐴)
3	1, 2	sscfn1 17089	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐴 Fn (dom dom 𝐴 × dom dom 𝐴))
4		eqidd 2824	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐵 = dom dom 𝐵)
5	1, 4	sscfn2 17090	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐵 Fn (dom dom 𝐵 × dom dom 𝐵))
6	3, 5, 1	ssc1 17093	. . 3 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐵)
7		simpr 487	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐵 ⊆cat 𝐶)
8		eqidd 2824	. . . . 5 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐶 = dom dom 𝐶)
9	7, 8	sscfn2 17090	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐶 Fn (dom dom 𝐶 × dom dom 𝐶))
10	5, 9, 7	ssc1 17093	. . 3 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐵 ⊆ dom dom 𝐶)
11	6, 10	sstrd 3979	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → dom dom 𝐴 ⊆ dom dom 𝐶)
12	3	adantr 483	. . . . 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 483	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐴 ⊆cat 𝐵)
14		simprl 769	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐴)
15		simprr 771	. . . . 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 17094	. . . 4 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐵𝑦))
17	5	adantr 483	. . . . 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 483	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝐵 ⊆cat 𝐶)
19	6	adantr 483	. . . . . 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 3970	. . . . 5 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → 𝑥 ∈ dom dom 𝐵)
21	19, 15	sseldd 3970	. . . . 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 17094	. . . 4 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐵𝑦) ⊆ (𝑥𝐶𝑦))
23	16, 22	sstrd 3979	. . 3 ⊢ (((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) ∧ (𝑥 ∈ dom dom 𝐴 ∧ 𝑦 ∈ dom dom 𝐴)) → (𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
24	23	ralrimivva 3193	. 2 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → ∀𝑥 ∈ dom dom 𝐴∀𝑦 ∈ dom dom 𝐴(𝑥𝐴𝑦) ⊆ (𝑥𝐶𝑦))
25		sscrel 17085	. . . . . 6 ⊢ Rel ⊆cat
26	25	brrelex2i 5611	. . . . 5 ⊢ (𝐵 ⊆cat 𝐶 → 𝐶 ∈ V)
27	26	adantl 484	. . . 4 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐶 ∈ V)
28		dmexg 7615	. . . 4 ⊢ (𝐶 ∈ V → dom 𝐶 ∈ V)
29		dmexg 7615	. . . 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 17092	. 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 711	1 ⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐴 ⊆cat 𝐶)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ⊆ wss 3938   class class class wbr 5068   × cxp 5555  dom cdm 5557   Fn wfn 6352  (class class class)co 7158   ⊆_cat cssc 17079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-ixp 8464  df-ssc 17082

This theorem is referenced by:  subsubc  17125