Theorem oppfvalg 49514

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 484	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
2	1	releqd 5738	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (Rel 𝑔 ↔ Rel 𝐺))
3	1	dmeqd 5864	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → dom 𝑔 = dom 𝐺)
4	3	releqd 5738	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (Rel dom 𝑔 ↔ Rel dom 𝐺))
5	2, 4	anbi12d 633	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((Rel 𝑔 ∧ Rel dom 𝑔) ↔ (Rel 𝐺 ∧ Rel dom 𝐺)))
6		simpl 482	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
7	1	tposeqd 8183	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → tpos 𝑔 = tpos 𝐺)
8	6, 7	opeq12d 4839	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ⟨𝑓, tpos 𝑔⟩ = ⟨𝐹, tpos 𝐺⟩)
9	5, 8	ifbieq1d 4506	. 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 49511	. 2 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
11		opex 5421	. . 3 ⊢ ⟨𝐹, tpos 𝐺⟩ ∈ V
12		0ex 5256	. . 3 ⊢ ∅ ∈ V
13	11, 12	ifex 4532	. 2 ⊢ if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ∈ V
14	9, 10, 13	ovmpoa 7525	1 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  ifcif 4481  ⟨cop 4588  dom cdm 5634  Rel wrel 5639  (class class class)co 7370  tpos ctpos 8179   oppFunc coppf 49510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6458  df-fun 6504  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-tpos 8180  df-oppf 49511

This theorem is referenced by:  oppfrcl3  49518  oppf1st2nd  49519  2oppf  49520  eloppf  49521  eloppf2  49522  oppfval  49524