Theorem oppf2 49803

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 49800	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
3		fvex 6895	. . . . 5 ⊢ (1st ‘𝐹) ∈ V
4		fvex 6895	. . . . . 6 ⊢ (2nd ‘𝐹) ∈ V
5	4	tposex 8256	. . . . 5 ⊢ tpos (2nd ‘𝐹) ∈ V
6	3, 5	op2ndd 7997	. . . 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 7428	. 2 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑀tpos (2nd ‘𝐹)𝑁))
9		ovtpos 8237	. 2 ⊢ (𝑀tpos (2nd ‘𝐹)𝑁) = (𝑁(2nd ‘𝐹)𝑀)
10	8, 9	eqtrdi 2820	1 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑁(2nd ‘𝐹)𝑀))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ⟨cop 4600  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  2^nd c2nd 7985  tpos ctpos 8221   Func cfunc 17911   oppFunc coppf 49785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-tpos 8222  df-map 8826  df-ixp 8896  df-func 17915  df-oppf 49786

This theorem is referenced by:  oppfdiag  50079