Theorem oppfvalg 49826

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 489	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
2	1	releqd 5768	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (Rel 𝑔 ↔ Rel 𝐺))
3	1	dmeqd 5898	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → dom 𝑔 = dom 𝐺)
4	3	releqd 5768	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (Rel dom 𝑔 ↔ Rel dom 𝐺))
5	2, 4	anbi12d 643	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((Rel 𝑔 ∧ Rel dom 𝑔) ↔ (Rel 𝐺 ∧ Rel dom 𝐺)))
6		simpl 487	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
7	1	tposeqd 8227	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → tpos 𝑔 = tpos 𝐺)
8	6, 7	opeq12d 4850	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ⟨𝑓, tpos 𝑔⟩ = ⟨𝐹, tpos 𝐺⟩)
9	5, 8	ifbieq1d 4517	. 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 49823	. 2 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
11		opex 5448	. . 3 ⊢ ⟨𝐹, tpos 𝐺⟩ ∈ V
12		0ex 5274	. . 3 ⊢ ∅ ∈ V
13	11, 12	ifex 4543	. 2 ⊢ if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ∈ V
14	9, 10, 13	ovmpoa 7568	1 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  ifcif 4492  ⟨cop 4600  dom cdm 5664  Rel wrel 5669  (class class class)co 7413  tpos ctpos 8223   oppFunc coppf 49822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5273  ax-pr 5407

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-res 5676  df-iota 6495  df-fun 6541  df-fv 6547  df-ov 7416  df-oprab 7417  df-mpo 7418  df-tpos 8224  df-oppf 49823

This theorem is referenced by:  oppfrcl3  49830  oppf1st2nd  49831  2oppf  49832  eloppf  49833  eloppf2  49834  oppfval  49836