Theorem ssc2 17997

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 2762	. . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽)
6	3, 5	sscfn2 17993	. . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7		sscrel 17988	. . . . . . 7 ⊢ Rel ⊆cat
8	7	brrelex2i 5708	. . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V)
9		dmexg 7913	. . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V)
10		dmexg 7913	. . . . . 6 ⊢ (dom 𝐽 ∈ V → dom dom 𝐽 ∈ V)
11	3, 8, 9, 10	4syl 20	. . . . 5 ⊢ (𝜑 → dom dom 𝐽 ∈ V)
12	4, 6, 11	isssc 17995	. . . 4 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
13	3, 12	mpbid 235	. . 3 ⊢ (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
14	13	simprd 501	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15		oveq1 7427	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16		oveq1 7427	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
17	15, 16	sseq12d 3964	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18		oveq2 7428	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19		oveq2 7428	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
20	18, 19	sseq12d 3964	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
21	17, 20	rspc2va 3588	. 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
22	1, 2, 14, 21	syl21anc 851	1 ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ⊆ wss 3899   class class class wbr 5103   × cxp 5649  dom cdm 5651   Fn wfn 6533  (class class class)co 7420   ⊆_cat cssc 17982

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-ixp 8926  df-ssc 17985

This theorem is used by:  ssctr  18000  ssceq  18001  subcss2  18018  iinfssc  50164  ssccatid  50179