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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 2779 | . . . 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 17003 | . . 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 6450 | . 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 6955 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | 12, 12 | xpex 7240 | . . . 4 ⊢ (𝐵 × 𝐵) ∈ V |
14 | 13, 12 | mpt2ex 7527 | . . 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 17002 | . . . 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 16463 | . . . 4 ⊢ comp = Slot (comp‘ndx) | |
17 | snsstp3 4580 | . . . 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 16303 | . . 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 2839 | 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 1601 ∈ wcel 2107 Vcvv 3398 ⦋csb 3751 {ctp 4402 〈cop 4404 ↦ cmpt 4965 × cxp 5353 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 1st c1st 7443 2nd c2nd 7444 1c1 10273 5c5 11433 ;cdc 11845 ndxcnx 16252 Basecbs 16255 Hom chom 16349 compcco 16350 Catccat 16710 Func cfunc 16899 Nat cnat 16986 FuncCat cfuc 16987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-hom 16362 df-cco 16363 df-fuc 16989 |
This theorem is referenced by: fucbas 17005 fuchom 17006 fucco 17007 |
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