postcof2 - Mathbox for Zhi Wang

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Theorem postcof2 50591

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⟩ curry_F (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
postcof1st.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
postcof1st.c	⊢ (𝜑 → 𝐶 ∈ Cat)
postcof1st.k	⊢ 𝐾 = ((1^st ‘ ⚬ )‘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(2^nd ‘𝐾)𝐻)‘𝐴)‘𝑋) = ((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘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⟩ curry_F (⟨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 = ((1^st ‘ ⚬ )‘𝐹)
7	1, 2, 3, 4, 5, 6	postcof2nd 50590	. . . 4 ⊢ (𝜑 → (2^nd ‘𝐾) = (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝑔)‘𝑥)(2^nd ‘𝐹)((1^st ‘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 7438	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (𝑔(𝐶 Nat 𝐷)ℎ) = (𝐺(𝐶 Nat 𝐷)𝐻))
11	8	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (1^st ‘𝑔) = (1^st ‘𝐺))
12	11	fveq1d 6887	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → ((1^st ‘𝑔)‘𝑥) = ((1^st ‘𝐺)‘𝑥))
13	9	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (1^st ‘ℎ) = (1^st ‘𝐻))
14	13	fveq1d 6887	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → ((1^st ‘ℎ)‘𝑥) = ((1^st ‘𝐻)‘𝑥))
15	12, 14	oveq12d 7438	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (((1^st ‘𝑔)‘𝑥)(2^nd ‘𝐹)((1^st ‘ℎ)‘𝑥)) = (((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥)))
16	15	fveq1d 6887	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → ((((1^st ‘𝑔)‘𝑥)(2^nd ‘𝐹)((1^st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)) = ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))
17	16	mpteq2dv 5198	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝑔)‘𝑥)(2^nd ‘𝐹)((1^st ‘ℎ)‘𝑥))‘(𝑎‘𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝑎‘𝑥))))
18	10, 17	mpteq12dv 5191	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ ℎ = 𝐻)) → (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝑔)‘𝑥)(2^nd ‘𝐹)((1^st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))) = (𝑎 ∈ (𝐺(𝐶 Nat 𝐷)𝐻) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))))
19		postcof2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
20		postcof2.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
21		ovex 7453	. . . . . 6 ⊢ (𝐺(𝐶 Nat 𝐷)𝐻) ∈ V
22	21	mptex 7229	. . . . 5 ⊢ (𝑎 ∈ (𝐺(𝐶 Nat 𝐷)𝐻) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))) ∈ V
23	22	a1i 11	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐺(𝐶 Nat 𝐷)𝐻) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))) ∈ V)
24	7, 18, 19, 20, 23	ovmpod 7572	. . 3 ⊢ (𝜑 → (𝐺(2^nd ‘𝐾)𝐻) = (𝑎 ∈ (𝐺(𝐶 Nat 𝐷)𝐻) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)))))
25		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
26	25	fveq1d 6887	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎‘𝑥) = (𝐴‘𝑥))
27	26	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝑎‘𝑥)) = ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥)))
28	27	mpteq2dv 5198	. . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝑎‘𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))))
29		postcof2.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))
30		fvex 6898	. . . . 5 ⊢ (Base‘𝐶) ∈ V
31	30	mptex 7229	. . . 4 ⊢ (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))) ∈ V
32	31	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))) ∈ V)
33	24, 28, 29, 32	fvmptd 7001	. 2 ⊢ (𝜑 → ((𝐺(2^nd ‘𝐾)𝐻)‘𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))))
34		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
35	34	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑋))
36	34	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐻)‘𝑥) = ((1^st ‘𝐻)‘𝑋))
37	35, 36	oveq12d 7438	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥)) = (((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑋)))
38	34	fveq2d 6889	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴‘𝑥) = (𝐴‘𝑋))
39	37, 38	fveq12d 6892	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥)) = ((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑋))‘(𝐴‘𝑋)))
40		postcof2.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
41		fvexd 6900	. 2 ⊢ (𝜑 → ((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑋))‘(𝐴‘𝑋)) ∈ V)
42	33, 39, 40, 41	fvmptd 7001	1 ⊢ (𝜑 → (((𝐺(2^nd ‘𝐾)𝐻)‘𝐴)‘𝑋) = ((((1^st ‘𝐺)‘𝑋)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑋))‘(𝐴‘𝑋)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⟨cop 4589 ↦ cmpt 5185 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 Basecbs 17389 Catccat 17840 Func cfunc 18031 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: fuconuniqlem 50872 fuconuniqvalslem 50874

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