Theorem eloppf2 49792

Description: Both components of a pre-image of a non-empty opposite functor exist; and the second component is a relation on triples. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
eloppf2.k	⊢ (𝐹 oppFunc 𝐺) = 𝐾
eloppf2.x	⊢ (𝜑 → 𝑋 ∈ 𝐾)

Assertion

Ref	Expression
eloppf2	⊢ (𝜑 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Rel 𝐺 ∧ Rel dom 𝐺)))

Proof of Theorem eloppf2

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

Step	Hyp	Ref	Expression
1		eloppf2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
2		eloppf2.k	. . . 4 ⊢ (𝐹 oppFunc 𝐺) = 𝐾
3	1, 2	eleqtrrdi 2880	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐹 oppFunc 𝐺))
4		df-oppf 49781	. . . 4 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
5	4	elmpocl 7649	. . 3 ⊢ (𝑋 ∈ (𝐹 oppFunc 𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
6	3, 5	syl 18	. 2 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
7		oppfvalg 49784	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
8	6, 7	syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
9	3, 8	eleqtrd 2871	. . . 4 ⊢ (𝜑 → 𝑋 ∈ if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
10	9	ne0d 4303	. . 3 ⊢ (𝜑 → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ≠ ∅)
11		iffalse 4498	. . . 4 ⊢ (¬ (Rel 𝐺 ∧ Rel dom 𝐺) → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) = ∅)
12	11	necon1ai 2991	. . 3 ⊢ (if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ≠ ∅ → (Rel 𝐺 ∧ Rel dom 𝐺))
13	10, 12	syl 18	. 2 ⊢ (𝜑 → (Rel 𝐺 ∧ Rel dom 𝐺))
14	6, 13	jca 520	1 ⊢ (𝜑 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Rel 𝐺 ∧ Rel dom 𝐺)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  ∅c0 4294  ifcif 4489  ⟨cop 4597  dom cdm 5659  Rel wrel 5664  (class class class)co 7408  tpos ctpos 8217   oppFunc coppf 49780

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 5258  ax-nul 5268  ax-pr 5402

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-ne 2965  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 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6490  df-fun 6536  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-tpos 8218  df-oppf 49781

This theorem is referenced by: (None)