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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 2732 | . . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
6 | 3, 5 | sscfn2 17747 | . . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
7 | sscrel 17742 | . . . . . . 7 ⊢ Rel ⊆cat | |
8 | 7 | brrelex2i 5725 | . . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V) |
9 | dmexg 7876 | . . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V) | |
10 | dmexg 7876 | . . . . . 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 17749 | . . . 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 496 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) |
15 | oveq1 7400 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦)) | |
16 | oveq1 7400 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦)) | |
17 | 15, 16 | sseq12d 4011 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦))) |
18 | oveq2 7401 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌)) | |
19 | oveq2 7401 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌)) | |
20 | 18, 19 | sseq12d 4011 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))) |
21 | 17, 20 | rspc2va 3619 | . 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)) |
22 | 1, 2, 14, 21 | syl21anc 836 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 Vcvv 3473 ⊆ wss 3944 class class class wbr 5141 × cxp 5667 dom cdm 5669 Fn wfn 6527 (class class class)co 7393 ⊆cat cssc 17736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-ixp 8875 df-ssc 17739 |
This theorem is referenced by: ssctr 17754 ssceq 17755 subcss2 17775 |
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