Theorem ssc2 17583

Description: Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
ssc2.1	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
ssc2.2	⊢ (𝜑 → 𝐻 ⊆cat 𝐽)
ssc2.3	⊢ (𝜑 → 𝑋 ∈ 𝑆)
ssc2.4	⊢ (𝜑 → 𝑌 ∈ 𝑆)

Assertion

Ref	Expression
ssc2	⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))

Proof of Theorem ssc2

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

Step	Hyp	Ref	Expression
1		ssc2.3	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
2		ssc2.4	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
3		ssc2.2	. . . 4 ⊢ (𝜑 → 𝐻 ⊆cat 𝐽)
4		ssc2.1	. . . . 5 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
5		eqidd 2737	. . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽)
6	3, 5	sscfn2 17579	. . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7		sscrel 17574	. . . . . . 7 ⊢ Rel ⊆cat
8	7	brrelex2i 5655	. . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V)
9		dmexg 7782	. . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V)
10		dmexg 7782	. . . . . 6 ⊢ (dom 𝐽 ∈ V → dom dom 𝐽 ∈ V)
11	3, 8, 9, 10	4syl 19	. . . . 5 ⊢ (𝜑 → dom dom 𝐽 ∈ V)
12	4, 6, 11	isssc 17581	. . . 4 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
13	3, 12	mpbid 231	. . 3 ⊢ (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
14	13	simprd 497	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15		oveq1 7314	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16		oveq1 7314	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
17	15, 16	sseq12d 3959	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18		oveq2 7315	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19		oveq2 7315	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
20	18, 19	sseq12d 3959	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
21	17, 20	rspc2va 3576	. 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
22	1, 2, 14, 21	syl21anc 836	1 ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062  Vcvv 3437   ⊆ wss 3892   class class class wbr 5081   × cxp 5598  dom cdm 5600   Fn wfn 6453  (class class class)co 7307   ⊆_cat cssc 17568

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-ixp 8717  df-ssc 17571

This theorem is referenced by:  ssctr  17586  ssceq  17587  subcss2  17607