Theorem postcofval 49839

Description: Value of the post-composition functor as a curry of the functor composition bifunctor. (Contributed by Zhi Wang, 11-Oct-2025.)

Hypotheses

Ref	Expression
postcofval.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
postcofval.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
postcofval.o	⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))
postcofval.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
postcofval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
postcofval.k	⊢ 𝐾 = ((1st ‘ ⚬ )‘𝐹)

Assertion

Ref	Expression
postcofval	⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))))⟩)

Distinct variable groups:   𝐶,𝑎,𝑔,ℎ,𝑥   𝐷,𝑎,𝑔,ℎ,𝑥   𝐸,𝑎,𝑔,ℎ,𝑥   𝐹,𝑎,𝑔,ℎ,𝑥   𝐾,𝑎,𝑔,ℎ   𝑄,𝑎,𝑔,ℎ   𝑅,𝑎,𝑔,ℎ   𝜑,𝑎,𝑔,ℎ,𝑥

Allowed substitution hints:   𝑄(𝑥)   𝑅(𝑥)   𝐾(𝑥)   ⚬ (𝑥,𝑔,ℎ,𝑎)

Proof of Theorem postcofval

Step	Hyp	Ref	Expression
1		postcofval.o	. . 3 ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))
2		postcofval.r	. . . 4 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
3	2	fucbas 17930	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘𝑅)
4		postcofval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5	4	func1st2nd 49551	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
6	5	funcrcl2 49554	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	5	funcrcl3 49555	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
8	2, 6, 7	fuccat 17940	. . 3 ⊢ (𝜑 → 𝑅 ∈ Cat)
9		postcofval.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
10		postcofval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
11	9, 10, 6	fuccat 17940	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
12	2, 9	oveq12i 7379	. . . 4 ⊢ (𝑅 ×c 𝑄) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
13		eqid 2736	. . . 4 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
14	12, 13, 10, 6, 7	fucofunca 49835	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ ((𝑅 ×c 𝑄) Func (𝐶 FuncCat 𝐸)))
15	9	fucbas 17930	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
16		postcofval.k	. . 3 ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝐹)
17		eqid 2736	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
18	9, 17	fuchom 17931	. . 3 ⊢ (𝐶 Nat 𝐷) = (Hom ‘𝑄)
19		eqid 2736	. . 3 ⊢ (Id‘𝑅) = (Id‘𝑅)
20	1, 3, 8, 11, 14, 15, 4, 16, 18, 19	curf1 18191	. 2 ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎)))⟩)
21		eqidd 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)))
22		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → 𝑔 ∈ (𝐶 Func 𝐷))
23	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → 𝐹 ∈ (𝐷 Func 𝐸))
24	21, 22, 23	fuco11b 49812	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔) = (𝐹 ∘func 𝑔))
25	24	mpteq2dva 5178	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔)) = (𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)))
26		eqidd 2737	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)))
27		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ))
28	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → 𝐹 ∈ (𝐷 Func 𝐸))
29	26, 19, 2, 27, 28	fucolid 49836	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥))))
30	29	mpteq2dva 5178	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎)) = (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))))
31	30	mpoeq3dv 7446	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎))) = (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥))))))
32	25, 31	opeq12d 4824	. 2 ⊢ (𝜑 → ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎)))⟩ = ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))))⟩)
33	20, 32	eqtrd 2771	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4573   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941  Basecbs 17179  Catccat 17630  Idccid 17631   Func cfunc 17821   ∘_func ccofu 17823   Nat cnat 17911   FuncCat cfuc 17912   ×_c cxpc 18134   curry_F ccurf 18176   ∘_F cfuco 49791

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17634  df-cid 17635  df-func 17825  df-cofu 17827  df-nat 17913  df-fuc 17914  df-xpc 18138  df-curf 18180  df-fuco 49792

This theorem is referenced by: (None)