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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 2726 | . . . 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 17952 | . . 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 6900 | . 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 7453 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | 12, 12 | xpex 7756 | . . . 4 ⊢ (𝐵 × 𝐵) ∈ V |
14 | 13, 12 | mpoex 8084 | . . 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 17951 | . . . 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 17398 | . . . 4 ⊢ comp = Slot (comp‘ndx) | |
17 | snsstp3 4823 | . . . 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 17176 | . . 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 2790 | 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 1533 ∈ wcel 2098 Vcvv 3461 ⦋csb 3889 {ctp 4634 〈cop 4636 ↦ cmpt 5232 × cxp 5676 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 1st c1st 7992 2nd c2nd 7993 1c1 11141 5c5 12303 ;cdc 12710 ndxcnx 17165 Basecbs 17183 Hom chom 17247 compcco 17248 Catccat 17647 Func cfunc 17843 Nat cnat 17934 FuncCat cfuc 17935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-hom 17260 df-cco 17261 df-fuc 17937 |
This theorem is referenced by: fucbas 17954 fuchom 17955 fuchomOLD 17956 fucco 17957 |
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