Theorem eloppf2 49133

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 2839	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐹 oppFunc 𝐺))
4		df-oppf 49122	. . . 4 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
5	4	elmpocl 7581	. . 3 ⊢ (𝑋 ∈ (𝐹 oppFunc 𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
6	3, 5	syl 17	. 2 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
7		oppfvalg 49125	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
9	3, 8	eleqtrd 2830	. . . 4 ⊢ (𝜑 → 𝑋 ∈ if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
10	9	ne0d 4289	. . 3 ⊢ (𝜑 → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ≠ ∅)
11		iffalse 4481	. . . 4 ⊢ (¬ (Rel 𝐺 ∧ Rel dom 𝐺) → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) = ∅)
12	11	necon1ai 2952	. . 3 ⊢ (if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ≠ ∅ → (Rel 𝐺 ∧ Rel dom 𝐺))
13	10, 12	syl 17	. 2 ⊢ (𝜑 → (Rel 𝐺 ∧ Rel dom 𝐺))
14	6, 13	jca 511	1 ⊢ (𝜑 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Rel 𝐺 ∧ Rel dom 𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3433  ∅c0 4280  ifcif 4472  ⟨cop 4579  dom cdm 5613  Rel wrel 5618  (class class class)co 7340  tpos ctpos 8149   oppFunc coppf 49121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pr 5367

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-res 5625  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7343  df-oprab 7344  df-mpo 7345  df-tpos 8150  df-oppf 49122

This theorem is referenced by: (None)