Theorem natoppf2 49262

Description: A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
natoppf.o	⊢ 𝑂 = (oppCat‘𝐶)
natoppf.p	⊢ 𝑃 = (oppCat‘𝐷)
natoppf.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
natoppf.m	⊢ 𝑀 = (𝑂 Nat 𝑃)
natoppfb.k	⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
natoppfb.l	⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
natoppf2.a	⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))

Assertion

Ref	Expression
natoppf2	⊢ (𝜑 → 𝐴 ∈ (𝐿𝑀𝐾))

Proof of Theorem natoppf2

Step	Hyp	Ref	Expression
1		natoppf.o	. . 3 ⊢ 𝑂 = (oppCat‘𝐶)
2		natoppf.p	. . 3 ⊢ 𝑃 = (oppCat‘𝐷)
3		natoppf.n	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		natoppf.m	. . 3 ⊢ 𝑀 = (𝑂 Nat 𝑃)
5		natoppf2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
6	3, 5	nat1st2nd 17856	. . 3 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
7	1, 2, 3, 4, 6	natoppf 49261	. 2 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐺), tpos (2nd ‘𝐺)⟩𝑀⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩))
8		natoppfb.l	. . . 4 ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
9	3	natrcl 17855	. . . . . 6 ⊢ (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
10	9	simprd 495	. . . . 5 ⊢ (𝐴 ∈ (𝐹𝑁𝐺) → 𝐺 ∈ (𝐶 Func 𝐷))
11		oppfval2 49169	. . . . 5 ⊢ (𝐺 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐺) = ⟨(1st ‘𝐺), tpos (2nd ‘𝐺)⟩)
12	5, 10, 11	3syl 18	. . . 4 ⊢ (𝜑 → ( oppFunc ‘𝐺) = ⟨(1st ‘𝐺), tpos (2nd ‘𝐺)⟩)
13	8, 12	eqtrd 2766	. . 3 ⊢ (𝜑 → 𝐿 = ⟨(1st ‘𝐺), tpos (2nd ‘𝐺)⟩)
14		natoppfb.k	. . . 4 ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
15	9	simpld 494	. . . . 5 ⊢ (𝐴 ∈ (𝐹𝑁𝐺) → 𝐹 ∈ (𝐶 Func 𝐷))
16		oppfval2 49169	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
17	5, 15, 16	3syl 18	. . . 4 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
18	14, 17	eqtrd 2766	. . 3 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
19	13, 18	oveq12d 7359	. 2 ⊢ (𝜑 → (𝐿𝑀𝐾) = (⟨(1st ‘𝐺), tpos (2nd ‘𝐺)⟩𝑀⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩))
20	7, 19	eleqtrrd 2834	1 ⊢ (𝜑 → 𝐴 ∈ (𝐿𝑀𝐾))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ⟨cop 4577  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  tpos ctpos 8150  oppCatcoppc 17612   Func cfunc 17756   Nat cnat 17846   oppFunc coppf 49154

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-map 8747  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-z 12464  df-dec 12584  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-hom 17180  df-cco 17181  df-cat 17569  df-cid 17570  df-oppc 17613  df-func 17760  df-nat 17848  df-oppf 49155

This theorem is referenced by:  natoppfb  49263