![]() |
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 | 5 | fndmd 6427 | . . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆)) |
7 | 6 | dmeqd 5738 | . . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆)) |
8 | dmxpid 5764 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
9 | 7, 8 | eqtr2di 2850 | . 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽) |
10 | 1, 2, 3, 4, 9 | issubc 17097 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 〈cop 4531 class class class wbr 5030 × cxp 5517 dom cdm 5519 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 compcco 16569 Catccat 16927 Idccid 16928 Homf chomf 16929 ⊆cat cssc 17069 Subcatcsubc 17071 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-pm 8392 df-ixp 8445 df-ssc 17072 df-subc 17074 |
This theorem is referenced by: 0subcat 17100 catsubcat 17101 subcidcl 17106 subccocl 17107 issubc3 17111 fullsubc 17112 rnghmsubcsetc 44601 rhmsubcsetc 44647 rhmsubcrngc 44653 srhmsubc 44700 rhmsubc 44714 srhmsubcALTV 44718 rhmsubcALTV 44732 |
Copyright terms: Public domain | W3C validator |