Theorem eloppf2 49912

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 2874	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐹 oppFunc 𝐺))
4		df-oppf 49901	. . . 4 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
5	4	elmpocl 7651	. . 3 ⊢ (𝑋 ∈ (𝐹 oppFunc 𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
6	3, 5	syl 18	. 2 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
7		oppfvalg 49904	. . . . . 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 2865	. . . 4 ⊢ (𝜑 → 𝑋 ∈ if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
10	9	ne0d 4295	. . 3 ⊢ (𝜑 → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ≠ ∅)
11		iffalse 4496	. . . 4 ⊢ (¬ (Rel 𝐺 ∧ Rel dom 𝐺) → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) = ∅)
12	11	necon1ai 2985	. . 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 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4286  ifcif 4487  ⟨cop 4595  dom cdm 5661  Rel wrel 5666  (class class class)co 7410  tpos ctpos 8217   oppFunc coppf 49900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-tpos 8218  df-oppf 49901

This theorem is referenced by: (None)