![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > issubc2 | Structured version Visualization version GIF version |
Description: Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
issubc.h | ⊢ 𝐻 = (Homf ‘𝐶) |
issubc.i | ⊢ 1 = (Id‘𝐶) |
issubc.o | ⊢ · = (comp‘𝐶) |
issubc.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
issubc2.a | ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
Ref | Expression |
---|---|
issubc2 | ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubc.h | . 2 ⊢ 𝐻 = (Homf ‘𝐶) | |
2 | issubc.i | . 2 ⊢ 1 = (Id‘𝐶) | |
3 | issubc.o | . 2 ⊢ · = (comp‘𝐶) | |
4 | issubc.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | issubc2.a | . . . . 5 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) | |
6 | fndm 6237 | . . . . 5 ⊢ (𝐽 Fn (𝑆 × 𝑆) → dom 𝐽 = (𝑆 × 𝑆)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆)) |
8 | 7 | dmeqd 5573 | . . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆)) |
9 | dmxpid 5592 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
10 | 8, 9 | syl6req 2831 | . 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽) |
11 | 1, 2, 3, 4, 10 | issubc 16891 | 1 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 〈cop 4404 class class class wbr 4888 × cxp 5355 dom cdm 5357 Fn wfn 6132 ‘cfv 6137 (class class class)co 6924 compcco 16361 Catccat 16721 Idccid 16722 Homf chomf 16723 ⊆cat cssc 16863 Subcatcsubc 16865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-pm 8145 df-ixp 8197 df-ssc 16866 df-subc 16868 |
This theorem is referenced by: 0subcat 16894 catsubcat 16895 subcidcl 16900 subccocl 16901 issubc3 16905 fullsubc 16906 rnghmsubcsetc 43006 rhmsubcsetc 43052 rhmsubcrngc 43058 srhmsubc 43105 rhmsubc 43119 srhmsubcALTV 43123 rhmsubcALTV 43137 |
Copyright terms: Public domain | W3C validator |