Theorem oppf2 49180

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 49177	. . . 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 8190	. . . . 5 ⊢ tpos (2nd ‘𝐹) ∈ V
6	3, 5	op2ndd 7932	. . . 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 7363	. 2 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑀tpos (2nd ‘𝐹)𝑁))
9		ovtpos 8171	. 2 ⊢ (𝑀tpos (2nd ‘𝐹)𝑁) = (𝑁(2nd ‘𝐹)𝑀)
10	8, 9	eqtrdi 2782	1 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑁(2nd ‘𝐹)𝑀))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ⟨cop 4579  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  tpos ctpos 8155   Func cfunc 17761   oppFunc coppf 49162

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 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3048  df-rex 3057  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-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-tpos 8156  df-map 8752  df-ixp 8822  df-func 17765  df-oppf 49163

This theorem is referenced by:  oppfdiag  49456