Theorem oppfval3 49936

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 6885	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐺, 𝐾⟩))
3		df-ov 7413	. . 3 ⊢ (𝐺 oppFunc 𝐾) = ( oppFunc ‘⟨𝐺, 𝐾⟩)
4	2, 3	eqtr4di 2816	. 2 ⊢ (𝜑 → ( oppFunc ‘𝐹) = (𝐺 oppFunc 𝐾))
5		oppfval3.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6	1, 5	eqeltrrd 2864	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐾⟩ ∈ (𝐶 Func 𝐷))
7		df-br 5110	. . . 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 49934	. . 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 2798	1 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨𝐺, tpos 𝐾⟩)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  tpos ctpos 8217   Func cfunc 17906   oppFunc coppf 49920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-tpos 8218  df-map 8822  df-ixp 8892  df-func 17910  df-oppf 49921

This theorem is referenced by:  cofuoppf  49948  oppc1stf  50086  oppc2ndf  50087