Theorem ssc2 16703

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 2761	. . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽)
6	3, 5	sscfn2 16699	. . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7		sscrel 16694	. . . . . . 7 ⊢ Rel ⊆cat
8	7	brrelex2i 5316	. . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V)
9		dmexg 7263	. . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V)
10		dmexg 7263	. . . . . 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 16701	. . . 4 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
13	3, 12	mpbid 222	. . 3 ⊢ (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
14	13	simprd 482	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15		oveq1 6821	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16		oveq1 6821	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
17	15, 16	sseq12d 3775	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18		oveq2 6822	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19		oveq2 6822	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
20	18, 19	sseq12d 3775	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
21	17, 20	rspc2va 3462	. 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
22	1, 2, 14, 21	syl21anc 1476	1 ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   ⊆ wss 3715   class class class wbr 4804   × cxp 5264  dom cdm 5266   Fn wfn 6044  (class class class)co 6814   ⊆_cat cssc 16688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-ixp 8077  df-ssc 16691

This theorem is referenced by:  ssctr  16706  ssceq  16707  subcss2  16724