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Mirrors > Home > MPE Home > Th. List > ssc1 | Structured version Visualization version GIF version |
Description: Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
isssc.1 | ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) |
isssc.2 | ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) |
ssc1.3 | ⊢ (𝜑 → 𝐻 ⊆cat 𝐽) |
Ref | Expression |
---|---|
ssc1 | ⊢ (𝜑 → 𝑆 ⊆ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssc1.3 | . . 3 ⊢ (𝜑 → 𝐻 ⊆cat 𝐽) | |
2 | isssc.1 | . . . 4 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
3 | isssc.2 | . . . 4 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) | |
4 | sscrel 17824 | . . . . . . 7 ⊢ Rel ⊆cat | |
5 | 4 | brrelex2i 5731 | . . . . . 6 ⊢ (𝐻 ⊆cat 𝐽 → 𝐽 ∈ V) |
6 | 1, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ V) |
7 | 3 | ssclem 17830 | . . . . 5 ⊢ (𝜑 → (𝐽 ∈ V ↔ 𝑇 ∈ V)) |
8 | 6, 7 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ V) |
9 | 2, 3, 8 | isssc 17831 | . . 3 ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))) |
10 | 1, 9 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))) |
11 | 10 | simpld 493 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ⊆ wss 3946 class class class wbr 5145 × cxp 5672 Fn wfn 6541 (class class class)co 7416 ⊆cat cssc 17818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-ixp 8919 df-ssc 17821 |
This theorem is referenced by: ssctr 17836 ssceq 17837 subcss1 17856 issubc3 17863 subsubc 17867 |
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