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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 2740 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
3 | 1, 2 | subcssc 17553 | . 2 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶)) |
4 | subcfn.2 | . 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽) | |
5 | 3, 4 | sscfn1 17527 | 1 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 × cxp 5588 dom cdm 5590 Fn wfn 6427 ‘cfv 6432 Homf chomf 17373 Subcatcsubc 17519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-pm 8601 df-ixp 8669 df-ssc 17520 df-subc 17522 |
This theorem is referenced by: subccat 17561 subsubc 17566 funcres 17609 funcres2 17611 idfusubc 45393 subthinc 46290 |
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