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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 6673 | . . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆)) |
| 7 | 6 | dmeqd 5916 | . . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆)) |
| 8 | dmxpid 5941 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
| 9 | 7, 8 | eqtr2di 2794 | . 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽) |
| 10 | 1, 2, 3, 4, 9 | issubc 17880 | 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 〈cop 4632 class class class wbr 5143 × cxp 5683 dom cdm 5685 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 compcco 17309 Catccat 17707 Idccid 17708 Homf chomf 17709 ⊆cat cssc 17851 Subcatcsubc 17853 |
| 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-oprab 7435 df-mpo 7436 df-pm 8869 df-ixp 8938 df-ssc 17854 df-subc 17856 |
| This theorem is referenced by: 0subcat 17883 catsubcat 17884 subcidcl 17889 subccocl 17890 issubc3 17894 fullsubc 17895 rnghmsubcsetc 20633 rhmsubcsetc 20662 rhmsubcrngc 20668 srhmsubc 20680 rhmsubc 20689 rhmsubcALTV 48201 srhmsubcALTV 48241 |
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