Theorem oppf2 49125

Description: Value of the morphism part of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)

Hypothesis

Ref	Expression
oppf1.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

Assertion

Ref	Expression
oppf2	⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑁(2nd ‘𝐹)𝑀))

Proof of Theorem oppf2

Step	Hyp	Ref	Expression
1		oppf1.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
2		oppfval2 49122	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
3		fvex 6835	. . . . 5 ⊢ (1st ‘𝐹) ∈ V
4		fvex 6835	. . . . . 6 ⊢ (2nd ‘𝐹) ∈ V
5	4	tposex 8193	. . . . 5 ⊢ tpos (2nd ‘𝐹) ∈ V
6	3, 5	op2ndd 7935	. . . 4 ⊢ (( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩ → (2nd ‘( oppFunc ‘𝐹)) = tpos (2nd ‘𝐹))
7	1, 2, 6	3syl 18	. . 3 ⊢ (𝜑 → (2nd ‘( oppFunc ‘𝐹)) = tpos (2nd ‘𝐹))
8	7	oveqd 7366	. 2 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑀tpos (2nd ‘𝐹)𝑁))
9		ovtpos 8174	. 2 ⊢ (𝑀tpos (2nd ‘𝐹)𝑁) = (𝑁(2nd ‘𝐹)𝑀)
10	8, 9	eqtrdi 2780	1 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑁(2nd ‘𝐹)𝑀))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  tpos ctpos 8158   Func cfunc 17761   oppFunc coppf 49107

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-tpos 8159  df-map 8755  df-ixp 8825  df-func 17765  df-oppf 49108

This theorem is referenced by:  oppfdiag  49401