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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 2738 | . . . . . 6 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
| 6 | 3, 5 | sscfn2 17862 | . . . . 5 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
| 7 | sscrel 17857 | . . . . . . 7 ⊢ Rel ⊆cat | |
| 8 | 7 | brrelex2i 5742 | . . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V) |
| 9 | dmexg 7923 | . . . . . 6 ⊢ (𝐽 ∈ V → dom 𝐽 ∈ V) | |
| 10 | dmexg 7923 | . . . . . 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 17864 | . . . 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 4017 | . . 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 4017 | . . 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 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 × cxp 5683 dom cdm 5685 Fn wfn 6556 (class class class)co 7431 ⊆cat cssc 17851 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-ixp 8938 df-ssc 17854 |
| This theorem is referenced by: ssctr 17869 ssceq 17870 subcss2 17888 |
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