Theorem oppfval3 49800

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

Hypotheses

Ref	Expression
oppfval3.g	⊢ (𝜑 → 𝐹 = ⟨𝐺, 𝐾⟩)
oppfval3.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

Assertion

Ref	Expression
oppfval3	⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨𝐺, tpos 𝐾⟩)

Proof of Theorem oppfval3

Step	Hyp	Ref	Expression
1		oppfval3.g	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨𝐺, 𝐾⟩)
2	1	fveq2d 6886	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐺, 𝐾⟩))
3		df-ov 7414	. . 3 ⊢ (𝐺 oppFunc 𝐾) = ( oppFunc ‘⟨𝐺, 𝐾⟩)
4	2, 3	eqtr4di 2822	. 2 ⊢ (𝜑 → ( oppFunc ‘𝐹) = (𝐺 oppFunc 𝐾))
5		oppfval3.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6	1, 5	eqeltrrd 2870	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐾⟩ ∈ (𝐶 Func 𝐷))
7		df-br 5114	. . . 4 ⊢ (𝐺(𝐶 Func 𝐷)𝐾 ↔ ⟨𝐺, 𝐾⟩ ∈ (𝐶 Func 𝐷))
8	6, 7	sylibr 237	. . 3 ⊢ (𝜑 → 𝐺(𝐶 Func 𝐷)𝐾)
9		oppfval 49798	. . 3 ⊢ (𝐺(𝐶 Func 𝐷)𝐾 → (𝐺 oppFunc 𝐾) = ⟨𝐺, tpos 𝐾⟩)
10	8, 9	syl 18	. 2 ⊢ (𝜑 → (𝐺 oppFunc 𝐾) = ⟨𝐺, tpos 𝐾⟩)
11	4, 10	eqtrd 2804	1 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨𝐺, tpos 𝐾⟩)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ⟨cop 4600   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  tpos ctpos 8220   Func cfunc 17910   oppFunc coppf 49784

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 7985  df-2nd 7986  df-tpos 8221  df-map 8825  df-ixp 8895  df-func 17914  df-oppf 49785

This theorem is referenced by:  cofuoppf  49812  oppc1stf  49950  oppc2ndf  49951