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| Mirrors > Home > MPE Home > Th. List > funcixp | Structured version Visualization version GIF version | ||
| Description: The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| funcixp.b | ⊢ 𝐵 = (Base‘𝐷) | 
| funcixp.h | ⊢ 𝐻 = (Hom ‘𝐷) | 
| funcixp.j | ⊢ 𝐽 = (Hom ‘𝐸) | 
| funcixp.f | ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | 
| Ref | Expression | 
|---|---|
| funcixp | ⊢ (𝜑 → 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | funcixp.f | . . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
| 2 | funcixp.b | . . . 4 ⊢ 𝐵 = (Base‘𝐷) | |
| 3 | eqid 2736 | . . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
| 4 | funcixp.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐷) | |
| 5 | funcixp.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐸) | |
| 6 | eqid 2736 | . . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷) | |
| 7 | eqid 2736 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
| 8 | eqid 2736 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
| 9 | eqid 2736 | . . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸) | |
| 10 | df-br 5143 | . . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
| 11 | 1, 10 | sylib 218 | . . . . . 6 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | 
| 12 | funcrcl 17909 | . . . . . 6 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
| 13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | 
| 14 | 13 | simpld 494 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) | 
| 15 | 13 | simprd 495 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat) | 
| 16 | 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | isfunc 17910 | . . 3 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))) | 
| 17 | 1, 16 | mpbid 232 | . 2 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))) | 
| 18 | 17 | simp2d 1143 | 1 ⊢ (𝜑 → 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 〈cop 4631 class class class wbr 5142 × cxp 5682 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 1st c1st 8013 2nd c2nd 8014 ↑m cmap 8867 Xcixp 8938 Basecbs 17248 Hom chom 17309 compcco 17310 Catccat 17708 Idccid 17709 Func cfunc 17900 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-ixp 8939 df-func 17904 | 
| This theorem is referenced by: funcf2 17914 funcfn2 17915 wunfunc 17947 | 
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