Theorem postcof2 50705

Description: A component of the action of a post-composition functor is obtained by applying the morphism part of 𝐹 to the component morphism. (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)
postcof2.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
postcof2.h	⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
postcof2.a	⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))
postcof2.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))

Assertion

Ref	Expression
postcof2	⊢ (𝜑 → (((𝐺(2nd ‘𝐾)𝐻)‘𝐴)‘𝑋) = ((((1st ‘𝐺)‘𝑋)(2nd ‘𝐹)((1st ‘𝐻)‘𝑋))‘(𝐴‘𝑋)))

Proof of Theorem postcof2

Dummy variables 𝑎 𝑔 ℎ 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		postcof1st.q	. . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		postcof1st.r	. . . . 5 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
3		postcof1st.o	. . . . 5 ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))
4		postcof1st.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5		postcof1st.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		postcof1st.k	. . . . 5 ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝐹)
7	1, 2, 3, 4, 5, 6	postcof2nd 50704	. . . 4 ⊢ (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))))))
8		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → 𝑔 = 𝐺)
9		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → ℎ = 𝐻)
10	8, 9	oveq12d 7461	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (𝑔(𝐶 Nat 𝐷)ℎ) = (𝐺(𝐶 Nat 𝐷)𝐻))
11	8	fveq2d 6912	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (1st ‘𝑔) = (1st ‘𝐺))
12	11	fveq1d 6910	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → ((1st ‘𝑔)‘𝑥) = ((1st ‘𝐺)‘𝑥))
13	9	fveq2d 6912	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (1st ‘ℎ) = (1st ‘𝐻))
14	13	fveq1d 6910	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → ((1st ‘ℎ)‘𝑥) = ((1st ‘𝐻)‘𝑥))
15	12, 14	oveq12d 7461	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥)) = (((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥)))
16	15	fveq1d 6910	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)) = ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))
17	16	mpteq2dv 5202	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝑎‘𝑥))))
18	10, 17	mpteq12dv 5195	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))) = (𝑎 ∈ (𝐺(𝐶 Nat 𝐷)𝐻) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))))
19		postcof2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
20		postcof2.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
21		ovex 7476	. . . . . 6 ⊢ (𝐺(𝐶 Nat 𝐷)𝐻) ∈ V
22	21	mptex 7252	. . . . 5 ⊢ (𝑎 ∈ (𝐺(𝐶 Nat 𝐷)𝐻) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))) ∈ V
23	22	a1i 11	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐺(𝐶 Nat 𝐷)𝐻) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))) ∈ V)
24	7, 18, 19, 20, 23	ovmpod 7596	. . 3 ⊢ (𝜑 → (𝐺(2nd ‘𝐾)𝐻) = (𝑎 ∈ (𝐺(𝐶 Nat 𝐷)𝐻) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))))
25		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
26	25	fveq1d 6910	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎‘𝑥) = (𝐴‘𝑥))
27	26	fveq2d 6912	. . . 4 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)) = ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝐴‘𝑥)))
28	27	mpteq2dv 5202	. . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝑎‘𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))))
29		postcof2.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))
30		fvex 6921	. . . . 5 ⊢ (Base‘𝐶) ∈ V
31	30	mptex 7252	. . . 4 ⊢ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))) ∈ V
32	31	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))) ∈ V)
33	24, 28, 29, 32	fvmptd 7024	. 2 ⊢ (𝜑 → ((𝐺(2nd ‘𝐾)𝐻)‘𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))))
34		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
35	34	fveq2d 6912	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑋))
36	34	fveq2d 6912	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1st ‘𝐻)‘𝑥) = ((1st ‘𝐻)‘𝑋))
37	35, 36	oveq12d 7461	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥)) = (((1st ‘𝐺)‘𝑋)(2nd ‘𝐹)((1st ‘𝐻)‘𝑋)))
38	34	fveq2d 6912	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴‘𝑥) = (𝐴‘𝑋))
39	37, 38	fveq12d 6915	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝐴‘𝑥)) = ((((1st ‘𝐺)‘𝑋)(2nd ‘𝐹)((1st ‘𝐻)‘𝑋))‘(𝐴‘𝑋)))
40		postcof2.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
41		fvexd 6923	. 2 ⊢ (𝜑 → ((((1st ‘𝐺)‘𝑋)(2nd ‘𝐹)((1st ‘𝐻)‘𝑋))‘(𝐴‘𝑋)) ∈ V)
42	33, 39, 40, 41	fvmptd 7024	1 ⊢ (𝜑 → (((𝐺(2nd ‘𝐾)𝐻)‘𝐴)‘𝑋) = ((((1st ‘𝐺)‘𝑋)(2nd ‘𝐹)((1st ‘𝐻)‘𝑋))‘(𝐴‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4838   ↦ cmpt 5189  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  Basecbs 17416  Catccat 17867   Func cfunc 18058   Nat cnat 18148   FuncCat cfuc 18149   curry_F ccurf 18413   ∘_F cfuco 50643

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-cat 17871  df-cid 17872  df-func 18062  df-cofu 18064  df-nat 18150  df-fuc 18151  df-xpc 18375  df-curf 18417  df-fuco 50644

This theorem is used by:  fuconuniqlem  50986  fuconuniqvalslem  50988