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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 17220 | . 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 6667 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋)) |
11 | 9 | fveq2d 6667 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑋)) |
12 | 10, 11 | opeq12d 4803 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) |
13 | 9 | fveq2d 6667 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐻)‘𝑥) = ((1st ‘𝐻)‘𝑋)) |
14 | 12, 13 | oveq12d 7163 | . . 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 6667 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋)) |
16 | 9 | fveq2d 6667 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘𝑥) = (𝑅‘𝑋)) |
17 | 14, 15, 16 | oveq123d 7166 | . 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 7180 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉 · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋)) ∈ V) | |
20 | 8, 17, 18, 19 | fvmptd 6767 | 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 1528 ∈ wcel 2105 Vcvv 3492 〈cop 4563 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 Basecbs 16471 compcco 16565 Nat cnat 17199 FuncCat cfuc 17200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-hom 16577 df-cco 16578 df-func 17116 df-nat 17201 df-fuc 17202 |
This theorem is referenced by: fuccocl 17222 fucass 17226 evlfcllem 17459 yonedalem3b 17517 |
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