natoppf2 - Mathbox for Zhi Wang

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Theorem natoppf2 49201

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 17922	. . 3 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
7	1, 2, 3, 4, 6	natoppf 49200	. 2 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩𝑀⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩))
8		natoppfb.l	. . . 4 ⊢ (𝜑 → 𝐿 = (oppFunc‘𝐺))
9	3	natrcl 17921	. . . . . 6 ⊢ (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
10	9	simprd 495	. . . . 5 ⊢ (𝐴 ∈ (𝐹𝑁𝐺) → 𝐺 ∈ (𝐶 Func 𝐷))
11		oppfval2 49114	. . . . 5 ⊢ (𝐺 ∈ (𝐶 Func 𝐷) → (oppFunc‘𝐺) = ⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩)
12	5, 10, 11	3syl 18	. . . 4 ⊢ (𝜑 → (oppFunc‘𝐺) = ⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩)
13	8, 12	eqtrd 2765	. . 3 ⊢ (𝜑 → 𝐿 = ⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩)
14		natoppfb.k	. . . 4 ⊢ (𝜑 → 𝐾 = (oppFunc‘𝐹))
15	9	simpld 494	. . . . 5 ⊢ (𝐴 ∈ (𝐹𝑁𝐺) → 𝐹 ∈ (𝐶 Func 𝐷))
16		oppfval2 49114	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (oppFunc‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
17	5, 15, 16	3syl 18	. . . 4 ⊢ (𝜑 → (oppFunc‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
18	14, 17	eqtrd 2765	. . 3 ⊢ (𝜑 → 𝐾 = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
19	13, 18	oveq12d 7407	. 2 ⊢ (𝜑 → (𝐿𝑀𝐾) = (⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩𝑀⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩))
20	7, 19	eleqtrrd 2832	1 ⊢ (𝜑 → 𝐴 ∈ (𝐿𝑀𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ⟨cop 4597 ‘cfv 6513 (class class class)co 7389 1^st c1st 7968 2^nd c2nd 7969 tpos ctpos 8206 oppCatcoppc 17678 Func cfunc 17822 Nat cnat 17912 oppFunccoppf 49099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-er 8673 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-hom 17250 df-cco 17251 df-cat 17635 df-cid 17636 df-oppc 17679 df-func 17826 df-nat 17914 df-oppf 49100

This theorem is referenced by: natoppfb 49202

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