Theorem oppfval3 49143

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 6830	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐺, 𝐾⟩))
3		df-ov 7356	. . 3 ⊢ (𝐺 oppFunc 𝐾) = ( oppFunc ‘⟨𝐺, 𝐾⟩)
4	2, 3	eqtr4di 2782	. 2 ⊢ (𝜑 → ( oppFunc ‘𝐹) = (𝐺 oppFunc 𝐾))
5		oppfval3.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6	1, 5	eqeltrrd 2829	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐾⟩ ∈ (𝐶 Func 𝐷))
7		df-br 5096	. . . 4 ⊢ (𝐺(𝐶 Func 𝐷)𝐾 ↔ ⟨𝐺, 𝐾⟩ ∈ (𝐶 Func 𝐷))
8	6, 7	sylibr 234	. . 3 ⊢ (𝜑 → 𝐺(𝐶 Func 𝐷)𝐾)
9		oppfval 49141	. . 3 ⊢ (𝐺(𝐶 Func 𝐷)𝐾 → (𝐺 oppFunc 𝐾) = ⟨𝐺, tpos 𝐾⟩)
10	8, 9	syl 17	. 2 ⊢ (𝜑 → (𝐺 oppFunc 𝐾) = ⟨𝐺, tpos 𝐾⟩)
11	4, 10	eqtrd 2764	1 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨𝐺, tpos 𝐾⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4585   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  tpos ctpos 8165   Func cfunc 17780   oppFunc coppf 49127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-tpos 8166  df-map 8762  df-ixp 8832  df-func 17784  df-oppf 49128

This theorem is referenced by:  cofuoppf  49155  oppc1stf  49293  oppc2ndf  49294