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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 2822 | . . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
6 | 3, 5 | sscfn2 17082 | . . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
7 | sscrel 17077 | . . . . . . 7 ⊢ Rel ⊆cat | |
8 | 7 | brrelex2i 5604 | . . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V) |
9 | dmexg 7607 | . . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V) | |
10 | dmexg 7607 | . . . . . 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 17084 | . . . 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 498 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) |
15 | oveq1 7157 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦)) | |
16 | oveq1 7157 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦)) | |
17 | 15, 16 | sseq12d 4000 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦))) |
18 | oveq2 7158 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌)) | |
19 | oveq2 7158 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌)) | |
20 | 18, 19 | sseq12d 4000 | . . 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 835 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 ⊆ wss 3936 class class class wbr 5059 × cxp 5548 dom cdm 5550 Fn wfn 6345 (class class class)co 7150 ⊆cat cssc 17071 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-ixp 8456 df-ssc 17074 |
This theorem is referenced by: ssctr 17089 ssceq 17090 subcss2 17107 |
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