prcoftposcurfucoa - Mathbox for Zhi Wang

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Theorem prcoftposcurfucoa 49571

Description: The pre-composition functor is the transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 2-Nov-2025.)

**Hypotheses**
Ref	Expression
prcoffunc.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
prcoffunc.e	⊢ (𝜑 → 𝐸 ∈ Cat)
prcoftposcurfuco.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
prcoftposcurfuco.o	⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curry_F ((⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∘_func (𝑄 swap_F 𝑅))))
prcoftposcurfucoa.m	⊢ (𝜑 → 𝑀 = ((1^st ‘ ⚬ )‘𝐹))
prcoftposcurfucoa.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
prcoftposcurfucoa	⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘_F 𝐹) = 𝑀)

**Proof of Theorem prcoftposcurfucoa**
Step	Hyp	Ref	Expression
1		relfunc 17784	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
2		prcoftposcurfucoa.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		1st2nd 7981	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 587	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5	4	oveq2d 7372	. 2 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘_F 𝐹) = (⟨𝐷, 𝐸⟩ −∘_F ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
6		prcoffunc.r	. . 3 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
7		prcoffunc.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
8		prcoftposcurfuco.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
9		prcoftposcurfuco.o	. . 3 ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curry_F ((⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∘_func (𝑄 swap_F 𝑅))))
10		prcoftposcurfucoa.m	. . . 4 ⊢ (𝜑 → 𝑀 = ((1^st ‘ ⚬ )‘𝐹))
11	4	fveq2d 6836	. . . 4 ⊢ (𝜑 → ((1^st ‘ ⚬ )‘𝐹) = ((1^st ‘ ⚬ )‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
12	10, 11	eqtrd 2769	. . 3 ⊢ (𝜑 → 𝑀 = ((1^st ‘ ⚬ )‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
13	2	func1st2nd 49263	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	6, 7, 8, 9, 12, 13	prcoftposcurfuco 49570	. 2 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘_F ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩) = 𝑀)
15	5, 14	eqtrd 2769	1 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘_F 𝐹) = 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ⟨cop 4584 Rel wrel 5627 ‘cfv 6490 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 Catccat 17585 Func cfunc 17776 ∘_func ccofu 17778 FuncCat cfuc 17867 curry_F ccurf 18131 swap_F cswapf 49446 ∘_F cfuco 49503 −∘_F cprcof 49560

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-fz 13422 df-struct 17072 df-slot 17107 df-ndx 17119 df-base 17135 df-hom 17199 df-cco 17200 df-cat 17589 df-cid 17590 df-func 17780 df-cofu 17782 df-nat 17868 df-fuc 17869 df-xpc 18093 df-curf 18135 df-swapf 49447 df-fuco 49504 df-prcof 49561

This theorem is referenced by: prcoffunca 49573

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