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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 2725 | . . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
6 | 3, 5 | sscfn2 17766 | . . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
7 | sscrel 17761 | . . . . . . 7 ⊢ Rel ⊆cat | |
8 | 7 | brrelex2i 5724 | . . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V) |
9 | dmexg 7888 | . . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V) | |
10 | dmexg 7888 | . . . . . 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 17768 | . . . 4 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))) |
13 | 3, 12 | mpbid 231 | . . 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 7409 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦)) | |
16 | oveq1 7409 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦)) | |
17 | 15, 16 | sseq12d 4008 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦))) |
18 | oveq2 7410 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌)) | |
19 | oveq2 7410 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌)) | |
20 | 18, 19 | sseq12d 4008 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))) |
21 | 17, 20 | rspc2va 3616 | . 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 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ⊆ wss 3941 class class class wbr 5139 × cxp 5665 dom cdm 5667 Fn wfn 6529 (class class class)co 7402 ⊆cat cssc 17755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-ixp 8889 df-ssc 17758 |
This theorem is referenced by: ssctr 17773 ssceq 17774 subcss2 17794 |
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