Theorem oppfvalg 49630

Description: Value of the opposite functor. (Contributed by Zhi Wang, 13-Nov-2025.)

Assertion

Ref	Expression
oppfvalg	⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))

Proof of Theorem oppfvalg

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

Step	Hyp	Ref	Expression
1		simpr 486	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
2	1	releqd 5725	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (Rel 𝑔 ↔ Rel 𝐺))
3	1	dmeqd 5854	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → dom 𝑔 = dom 𝐺)
4	3	releqd 5725	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (Rel dom 𝑔 ↔ Rel dom 𝐺))
5	2, 4	anbi12d 639	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((Rel 𝑔 ∧ Rel dom 𝑔) ↔ (Rel 𝐺 ∧ Rel dom 𝐺)))
6		simpl 484	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
7	1	tposeqd 8173	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → tpos 𝑔 = tpos 𝐺)
8	6, 7	opeq12d 4815	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ⟨𝑓, tpos 𝑔⟩ = ⟨𝐹, tpos 𝐺⟩)
9	5, 8	ifbieq1d 4482	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
10		df-oppf 49627	. 2 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
11		opex 5406	. . 3 ⊢ ⟨𝐹, tpos 𝐺⟩ ∈ V
12		0ex 5232	. . 3 ⊢ ∅ ∈ V
13	11, 12	ifex 4508	. 2 ⊢ if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ∈ V
14	9, 10, 13	ovmpoa 7515	1 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433  ∅c0 4264  ifcif 4457  ⟨cop 4564  dom cdm 5621  Rel wrel 5626  (class class class)co 7360  tpos ctpos 8169   oppFunc coppf 49626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-res 5633  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-tpos 8170  df-oppf 49627

This theorem is referenced by:  oppfrcl3  49634  oppf1st2nd  49635  2oppf  49636  eloppf  49637  eloppf2  49638  oppfval  49640