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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 17925 | . 2 ⊢ (𝜑 → (𝑆(〈𝐹, 𝐺〉 ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)))) |
9 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
10 | 9 | fveq2d 6895 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋)) |
11 | 9 | fveq2d 6895 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑋)) |
12 | 10, 11 | opeq12d 4881 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
13 | 9 | fveq2d 6895 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐻)‘𝑥) = ((1st ‘𝐻)‘𝑋)) |
14 | 12, 13 | oveq12d 7430 | . . 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 6895 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋)) |
16 | 9 | fveq2d 6895 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘𝑥) = (𝑅‘𝑋)) |
17 | 14, 15, 16 | oveq123d 7433 | . 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 7447 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋)) ∈ V) | |
20 | 8, 17, 18, 19 | fvmptd 7005 | 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 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 〈cop 4634 ‘cfv 6543 (class class class)co 7412 1st c1st 7977 Basecbs 17151 compcco 17216 Nat cnat 17902 FuncCat cfuc 17903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-hom 17228 df-cco 17229 df-func 17815 df-nat 17904 df-fuc 17905 |
This theorem is referenced by: fuccocl 17927 fucass 17931 evlfcllem 18184 yonedalem3b 18242 |
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