Theorem oppfval3 49170

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 6821	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐺, 𝐾⟩))
3		df-ov 7344	. . 3 ⊢ (𝐺 oppFunc 𝐾) = ( oppFunc ‘⟨𝐺, 𝐾⟩)
4	2, 3	eqtr4di 2784	. 2 ⊢ (𝜑 → ( oppFunc ‘𝐹) = (𝐺 oppFunc 𝐾))
5		oppfval3.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6	1, 5	eqeltrrd 2832	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐾⟩ ∈ (𝐶 Func 𝐷))
7		df-br 5087	. . . 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 49168	. . 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 2766	1 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨𝐺, tpos 𝐾⟩)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ⟨cop 4577   class class class wbr 5086  ‘cfv 6476  (class class class)co 7341  tpos ctpos 8150   Func cfunc 17756   oppFunc coppf 49154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-tpos 8151  df-map 8747  df-ixp 8817  df-func 17760  df-oppf 49155

This theorem is referenced by:  cofuoppf  49182  oppc1stf  49320  oppc2ndf  49321