Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fuccoval | Structured version Visualization version GIF version |
Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
fucco.q | ⊢ 𝑄 = (𝐶 FuncCat 𝐷) |
fucco.n | ⊢ 𝑁 = (𝐶 Nat 𝐷) |
fucco.a | ⊢ 𝐴 = (Base‘𝐶) |
fucco.o | ⊢ · = (comp‘𝐷) |
fucco.x | ⊢ ∙ = (comp‘𝑄) |
fucco.f | ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺)) |
fucco.g | ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻)) |
fuccoval.f | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
fuccoval | ⊢ (𝜑 → ((𝑆(〈𝐹, 𝐺〉 ∙ 𝐻)𝑅)‘𝑋) = ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fucco.q | . . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷) | |
2 | fucco.n | . . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
3 | fucco.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
4 | fucco.o | . . 3 ⊢ · = (comp‘𝐷) | |
5 | fucco.x | . . 3 ⊢ ∙ = (comp‘𝑄) | |
6 | fucco.f | . . 3 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺)) | |
7 | fucco.g | . . 3 ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | fucco 17680 | . 2 ⊢ (𝜑 → (𝑆(〈𝐹, 𝐺〉 ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)))) |
9 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
10 | 9 | fveq2d 6778 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋)) |
11 | 9 | fveq2d 6778 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑋)) |
12 | 10, 11 | opeq12d 4812 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
13 | 9 | fveq2d 6778 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐻)‘𝑥) = ((1st ‘𝐻)‘𝑋)) |
14 | 12, 13 | oveq12d 7293 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐻)‘𝑥)) = (〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))) |
15 | 9 | fveq2d 6778 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋)) |
16 | 9 | fveq2d 6778 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘𝑥) = (𝑅‘𝑋)) |
17 | 14, 15, 16 | oveq123d 7296 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑆‘𝑥)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)) = ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋))) |
18 | fuccoval.f | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | ovexd 7310 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋)) ∈ V) | |
20 | 8, 17, 18, 19 | fvmptd 6882 | 1 ⊢ (𝜑 → ((𝑆(〈𝐹, 𝐺〉 ∙ 𝐻)𝑅)‘𝑋) = ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 〈cop 4567 ‘cfv 6433 (class class class)co 7275 1st c1st 7829 Basecbs 16912 compcco 16974 Nat cnat 17657 FuncCat cfuc 17658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-hom 16986 df-cco 16987 df-func 17573 df-nat 17659 df-fuc 17660 |
This theorem is referenced by: fuccocl 17682 fucass 17686 evlfcllem 17939 yonedalem3b 17997 |
Copyright terms: Public domain | W3C validator |