Theorem postcof2nd 50590

Description: The morphism part of the explicit post-composition functor sends a pair of functors to the induced map on natural transformations. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
postcof1st.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
postcof1st.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
postcof1st.o	⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))
postcof1st.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
postcof1st.c	⊢ (𝜑 → 𝐶 ∈ Cat)
postcof1st.k	⊢ 𝐾 = ((1st ‘ ⚬ )‘𝐹)

Assertion

Ref	Expression
postcof2nd	⊢ (𝜑 → (2nd ‘𝐾) = (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   𝜑,𝑎,𝑔,ℎ,𝑥   𝐾,𝑎,𝑔,ℎ   𝑄,𝑎,𝑔,ℎ   𝑅,𝑎,𝑔,ℎ

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

Proof of Theorem postcof2nd

Step	Hyp	Ref	Expression
1		postcof1st.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		postcof1st.r	. . 3 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
3		postcof1st.o	. . 3 ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))
4		postcof1st.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5		postcof1st.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		postcof1st.k	. . 3 ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝐹)
7	1, 2, 3, 4, 5, 6	postcofval 50587	. 2 ⊢ (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)))))⟩)
8		ovex 7453	. . . 4 ⊢ (𝐶 Func 𝐷) ∈ V
9	8	mptex 7229	. . 3 ⊢ (𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)) ∈ V
10	8, 8	mpoex 8092	. . 3 ⊢ (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))))) ∈ V
11	9, 10	op2ndd 8012	. 2 ⊢ (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)))))⟩ → (2nd ‘𝐾) = (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))))))
12	7, 11	syl 18	1 ⊢ (𝜑 → (2nd ‘𝐾) = (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))))))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ⟨cop 4589   ↦ cmpt 5185  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17389  Catccat 17840   Func cfunc 18031   ∘_func ccofu 18033   Nat cnat 18121   FuncCat cfuc 18122   curry_F ccurf 18386   ∘_F cfuco 50529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11258  ax-resscn 11259  ax-1cn 11260  ax-icn 11261  ax-addcl 11262  ax-addrcl 11263  ax-mulcl 11264  ax-mulrcl 11265  ax-mulcom 11266  ax-addass 11267  ax-mulass 11268  ax-distr 11269  ax-i2m1 11270  ax-1ne0 11271  ax-1rid 11272  ax-rnegex 11273  ax-rrecex 11274  ax-cnre 11275  ax-pre-lttri 11276  ax-pre-lttrn 11277  ax-pre-ltadd 11278  ax-pre-mulgt0 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8301  df-wrecs 8332  df-recs 8381  df-rdg 8420  df-1o 8478  df-er 8719  df-map 8851  df-ixp 8928  df-en 8976  df-dom 8977  df-sdom 8978  df-fin 8979  df-pnf 11347  df-mnf 11348  df-xr 11349  df-ltxr 11350  df-le 11351  df-sub 11545  df-neg 11546  df-nn 12338  df-2 12407  df-3 12408  df-4 12409  df-5 12410  df-6 12411  df-7 12412  df-8 12413  df-9 12414  df-n0 12609  df-z 12696  df-dec 12817  df-uz 12968  df-fz 13642  df-struct 17327  df-slot 17362  df-ndx 17374  df-base 17390  df-hom 17454  df-cco 17455  df-cat 17844  df-cid 17845  df-func 18035  df-cofu 18037  df-nat 18123  df-fuc 18124  df-xpc 18348  df-curf 18390  df-fuco 50530

This theorem is used by:  postcof2  50591