Theorem ssc2 17855

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 2763	. . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽)
6	3, 5	sscfn2 17851	. . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7		sscrel 17846	. . . . . . 7 ⊢ Rel ⊆cat
8	7	brrelex2i 5704	. . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V)
9		dmexg 7882	. . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V)
10		dmexg 7882	. . . . . 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 17853	. . . 4 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
13	3, 12	mpbid 234	. . 3 ⊢ (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
14	13	simprd 499	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15		oveq1 7403	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16		oveq1 7403	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
17	15, 16	sseq12d 3969	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18		oveq2 7404	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19		oveq2 7404	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
20	18, 19	sseq12d 3969	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
21	17, 20	rspc2va 3593	. 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
22	1, 2, 14, 21	syl21anc 848	1 ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  Vcvv 3454   ⊆ wss 3904   class class class wbr 5100   × cxp 5645  dom cdm 5647   Fn wfn 6516  (class class class)co 7396   ⊆_cat cssc 17840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-ixp 8880  df-ssc 17843

This theorem is referenced by:  ssctr  17858  ssceq  17859  subcss2  17876  iinfssc  49678  ssccatid  49693