Theorem oppfval 48952

Description: Value of the opposite functor. (Contributed by Zhi Wang, 4-Nov-2025.)

Assertion

Ref	Expression
oppfval	⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹oppFunc𝐺) = ⟨𝐹, tpos 𝐺⟩)

Proof of Theorem oppfval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17862	. . 3 ⊢ Rel (𝐶 Func 𝐷)
2	1	brrelex12i 5707	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
3		simpl 482	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
4		simpr 484	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
5	4	tposeqd 8223	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → tpos 𝑔 = tpos 𝐺)
6	3, 5	opeq12d 4855	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ⟨𝑓, tpos 𝑔⟩ = ⟨𝐹, tpos 𝐺⟩)
7		df-oppf 48951	. . 3 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ ⟨𝑓, tpos 𝑔⟩)
8		opex 5437	. . 3 ⊢ ⟨𝐹, tpos 𝐺⟩ ∈ V
9	6, 7, 8	ovmpoa 7557	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹oppFunc𝐺) = ⟨𝐹, tpos 𝐺⟩)
10	2, 9	syl 17	1 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹oppFunc𝐺) = ⟨𝐹, tpos 𝐺⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3457  ⟨cop 4605   class class class wbr 5117  (class class class)co 7400  tpos ctpos 8219   Func cfunc 17854  oppFunccoppf 48950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-tpos 8220  df-func 17858  df-oppf 48951

This theorem is referenced by:  oppfoppc  48953  ranval  49356