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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 17330 | . 2 ⊢ (𝜑 → (𝑆(〈𝐹, 𝐺〉 ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)))) |
9 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
10 | 9 | fveq2d 6672 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋)) |
11 | 9 | fveq2d 6672 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑋)) |
12 | 10, 11 | opeq12d 4766 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
13 | 9 | fveq2d 6672 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐻)‘𝑥) = ((1st ‘𝐻)‘𝑋)) |
14 | 12, 13 | oveq12d 7182 | . . 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 6672 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋)) |
16 | 9 | fveq2d 6672 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘𝑥) = (𝑅‘𝑋)) |
17 | 14, 15, 16 | oveq123d 7185 | . 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 7199 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋)) ∈ V) | |
20 | 8, 17, 18, 19 | fvmptd 6776 | 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 399 = wceq 1542 ∈ wcel 2113 Vcvv 3397 〈cop 4519 ‘cfv 6333 (class class class)co 7164 1st c1st 7705 Basecbs 16579 compcco 16673 Nat cnat 17309 FuncCat cfuc 17310 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-hom 16685 df-cco 16686 df-func 17226 df-nat 17311 df-fuc 17312 |
This theorem is referenced by: fuccocl 17332 fucass 17336 evlfcllem 17580 yonedalem3b 17638 |
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