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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 | fndm 5990 | . . . . 5 ⊢ (𝐽 Fn (𝑆 × 𝑆) → dom 𝐽 = (𝑆 × 𝑆)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆)) |
8 | 7 | dmeqd 5326 | . . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆)) |
9 | dmxpid 5345 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
10 | 8, 9 | syl6req 2673 | . 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽) |
11 | 1, 2, 3, 4, 10 | issubc 16495 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 〈cop 4183 class class class wbr 4653 × cxp 5112 dom cdm 5114 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 compcco 15953 Catccat 16325 Idccid 16326 Homf chomf 16327 ⊆cat cssc 16467 Subcatcsubc 16469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pm 7860 df-ixp 7909 df-ssc 16470 df-subc 16472 |
This theorem is referenced by: 0subcat 16498 catsubcat 16499 subcidcl 16504 subccocl 16505 issubc3 16509 fullsubc 16510 rnghmsubcsetc 41977 rhmsubcsetc 42023 rhmsubcrngc 42029 srhmsubc 42076 rhmsubc 42090 srhmsubcALTV 42094 rhmsubcALTV 42108 |
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