Theorem ssc2 17784

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 2730	. . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽)
6	3, 5	sscfn2 17780	. . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7		sscrel 17775	. . . . . . 7 ⊢ Rel ⊆cat
8	7	brrelex2i 5695	. . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V)
9		dmexg 7877	. . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V)
10		dmexg 7877	. . . . . 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 17782	. . . 4 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
13	3, 12	mpbid 232	. . 3 ⊢ (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
14	13	simprd 495	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15		oveq1 7394	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16		oveq1 7394	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
17	15, 16	sseq12d 3980	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18		oveq2 7395	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19		oveq2 7395	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
20	18, 19	sseq12d 3980	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
21	17, 20	rspc2va 3600	. 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
22	1, 2, 14, 21	syl21anc 837	1 ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ 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:  ssctr  17787  ssceq  17788  subcss2  17805  iinfssc  49046  ssccatid  49061