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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 6651 | . . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆)) |
7 | 6 | dmeqd 5903 | . . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆)) |
8 | dmxpid 5927 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
9 | 7, 8 | eqtr2di 2790 | . 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽) |
10 | 1, 2, 3, 4, 9 | issubc 17781 | 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 〈cop 4633 class class class wbr 5147 × cxp 5673 dom cdm 5675 Fn wfn 6535 ‘cfv 6540 (class class class)co 7404 compcco 17205 Catccat 17604 Idccid 17605 Homf chomf 17606 ⊆cat cssc 17750 Subcatcsubc 17752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-pm 8819 df-ixp 8888 df-ssc 17753 df-subc 17755 |
This theorem is referenced by: 0subcat 17784 catsubcat 17785 subcidcl 17790 subccocl 17791 issubc3 17795 fullsubc 17796 rnghmsubcsetc 46777 rhmsubcsetc 46823 rhmsubcrngc 46829 srhmsubc 46876 rhmsubc 46890 srhmsubcALTV 46894 rhmsubcALTV 46908 |
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