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Mirrors > Home > MPE Home > Th. List > fuccofval | Structured version Visualization version GIF version |
Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
fucval.q | ⊢ 𝑄 = (𝐶 FuncCat 𝐷) |
fucval.b | ⊢ 𝐵 = (𝐶 Func 𝐷) |
fucval.n | ⊢ 𝑁 = (𝐶 Nat 𝐷) |
fucval.a | ⊢ 𝐴 = (Base‘𝐶) |
fucval.o | ⊢ · = (comp‘𝐷) |
fucval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
fucval.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
fuccofval.x | ⊢ ∙ = (comp‘𝑄) |
Ref | Expression |
---|---|
fuccofval | ⊢ (𝜑 → ∙ = (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fucval.q | . . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷) | |
2 | fucval.b | . . . 4 ⊢ 𝐵 = (𝐶 Func 𝐷) | |
3 | fucval.n | . . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
4 | fucval.a | . . . 4 ⊢ 𝐴 = (Base‘𝐶) | |
5 | fucval.o | . . . 4 ⊢ · = (comp‘𝐷) | |
6 | fucval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
7 | fucval.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
8 | eqidd 2799 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | fucval 17220 | . . 3 ⊢ (𝜑 → 𝑄 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉}) |
10 | 9 | fveq2d 6649 | . 2 ⊢ (𝜑 → (comp‘𝑄) = (comp‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉})) |
11 | fuccofval.x | . 2 ⊢ ∙ = (comp‘𝑄) | |
12 | 2 | ovexi 7169 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | 12, 12 | xpex 7456 | . . . 4 ⊢ (𝐵 × 𝐵) ∈ V |
14 | 13, 12 | mpoex 7760 | . . 3 ⊢ (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) ∈ V |
15 | catstr 17219 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉} Struct 〈1, ;15〉 | |
16 | ccoid 16682 | . . . 4 ⊢ comp = Slot (comp‘ndx) | |
17 | snsstp3 4711 | . . . 4 ⊢ {〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉} ⊆ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉} | |
18 | 15, 16, 17 | strfv 16523 | . . 3 ⊢ ((𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) ∈ V → (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (comp‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉})) |
19 | 14, 18 | ax-mp 5 | . 2 ⊢ (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (comp‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝑁〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉}) |
20 | 10, 11, 19 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → ∙ = (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉 · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⦋csb 3828 {ctp 4529 〈cop 4531 ↦ cmpt 5110 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1st c1st 7669 2nd c2nd 7670 1c1 10527 5c5 11683 ;cdc 12086 ndxcnx 16472 Basecbs 16475 Hom chom 16568 compcco 16569 Catccat 16927 Func cfunc 17116 Nat cnat 17203 FuncCat cfuc 17204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-hom 16581 df-cco 16582 df-fuc 17206 |
This theorem is referenced by: fucbas 17222 fuchom 17223 fucco 17224 |
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