Theorem oppfval3 50393

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 6897	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐺, 𝐾⟩))
3		df-ov 7430	. . 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 5106	. . . 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 50391	. . 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⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  ⟨cop 4837   class class class wbr 5105  ‘cfv 6546  (class class class)co 7427  tpos ctpos 8253   Func cfunc 18041   oppFunc coppf 50377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-tpos 8254  df-map 8860  df-ixp 8937  df-func 18045  df-oppf 50378

This theorem is used by:  cofuoppf  50405  oppc1stf  50549  oppc2ndf  50550