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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 2771 | . . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
4 | funcixp.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐷) | |
5 | funcixp.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐸) | |
6 | eqid 2771 | . . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷) | |
7 | eqid 2771 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
8 | eqid 2771 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
9 | eqid 2771 | . . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸) | |
10 | df-br 4787 | . . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
11 | 1, 10 | sylib 208 | . . . . . 6 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
12 | funcrcl 16726 | . . . . . 6 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
14 | 13 | simpld 482 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) |
15 | 13 | simprd 483 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat) |
16 | 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | isfunc 16727 | . . 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 222 | . 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 1137 | 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 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∀wral 3061 〈cop 4322 class class class wbr 4786 × cxp 5247 ⟶wf 6025 ‘cfv 6029 (class class class)co 6792 1st c1st 7313 2nd c2nd 7314 ↑𝑚 cmap 8009 Xcixp 8062 Basecbs 16060 Hom chom 16156 compcco 16157 Catccat 16528 Idccid 16529 Func cfunc 16717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-map 8011 df-ixp 8063 df-func 16721 |
This theorem is referenced by: funcf2 16731 funcfn2 16732 wunfunc 16762 |
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