Theorem oppf2 50587

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 50584	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
3		fvex 6929	. . . . 5 ⊢ (1st ‘𝐹) ∈ V
4		fvex 6929	. . . . . 6 ⊢ (2nd ‘𝐹) ∈ V
5	4	tposex 8315	. . . . 5 ⊢ tpos (2nd ‘𝐹) ∈ V
6	3, 5	op2ndd 8046	. . . 4 ⊢ (( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩ → (2nd ‘( oppFunc ‘𝐹)) = tpos (2nd ‘𝐹))
7	1, 2, 6	3syl 19	. . 3 ⊢ (𝜑 → (2nd ‘( oppFunc ‘𝐹)) = tpos (2nd ‘𝐹))
8	7	oveqd 7468	. 2 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑀tpos (2nd ‘𝐹)𝑁))
9		ovtpos 8296	. 2 ⊢ (𝑀tpos (2nd ‘𝐹)𝑁) = (𝑁(2nd ‘𝐹)𝑀)
10	8, 9	eqtrdi 2814	1 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑁(2nd ‘𝐹)𝑀))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ⟨cop 4840  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  tpos ctpos 8280   Func cfunc 18079   oppFunc coppf 50569

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-tpos 8281  df-map 8887  df-ixp 8964  df-func 18083  df-oppf 50570

This theorem is used by:  oppfdiag  50882