![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subcfn | Structured version Visualization version GIF version |
Description: An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
subcixp.1 | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
subcfn.2 | ⊢ (𝜑 → 𝑆 = dom dom 𝐽) |
Ref | Expression |
---|---|
subcfn | ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcixp.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
2 | eqid 2726 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
3 | 1, 2 | subcssc 17852 | . 2 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶)) |
4 | subcfn.2 | . 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽) | |
5 | 3, 4 | sscfn1 17826 | 1 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 × cxp 5671 dom cdm 5673 Fn wfn 6539 ‘cfv 6544 Homf chomf 17672 Subcatcsubc 17818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-pm 8848 df-ixp 8917 df-ssc 17819 df-subc 17821 |
This theorem is referenced by: subccat 17860 subsubc 17865 funcres 17908 funcres2 17910 idfusubc 17912 subthinc 48395 |
Copyright terms: Public domain | W3C validator |