![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ssc2 | Structured version Visualization version GIF version |
Description: Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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) |
Ref | Expression |
---|---|
ssc2 | ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)) |
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 2736 | . . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
6 | 3, 5 | sscfn2 17866 | . . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
7 | sscrel 17861 | . . . . . . 7 ⊢ Rel ⊆cat | |
8 | 7 | brrelex2i 5746 | . . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V) |
9 | dmexg 7924 | . . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V) | |
10 | dmexg 7924 | . . . . . 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 17868 | . . . 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 7438 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦)) | |
16 | oveq1 7438 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦)) | |
17 | 15, 16 | sseq12d 4029 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦))) |
18 | oveq2 7439 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌)) | |
19 | oveq2 7439 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌)) | |
20 | 18, 19 | sseq12d 4029 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))) |
21 | 17, 20 | rspc2va 3634 | . 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)) |
22 | 1, 2, 14, 21 | syl21anc 838 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 × cxp 5687 dom cdm 5689 Fn wfn 6558 (class class class)co 7431 ⊆cat cssc 17855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-ixp 8937 df-ssc 17858 |
This theorem is referenced by: ssctr 17873 ssceq 17874 subcss2 17894 |
Copyright terms: Public domain | W3C validator |