Theorem eloppf2 49493

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 2848	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐹 oppFunc 𝐺))
4		df-oppf 49482	. . . 4 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
5	4	elmpocl 7609	. . 3 ⊢ (𝑋 ∈ (𝐹 oppFunc 𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
6	3, 5	syl 17	. 2 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
7		oppfvalg 49485	. . . . . 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 2839	. . . 4 ⊢ (𝜑 → 𝑋 ∈ if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
10	9	ne0d 4296	. . 3 ⊢ (𝜑 → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ≠ ∅)
11		iffalse 4490	. . . 4 ⊢ (¬ (Rel 𝐺 ∧ Rel dom 𝐺) → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) = ∅)
12	11	necon1ai 2960	. . 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 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442  ∅c0 4287  ifcif 4481  ⟨cop 4588  dom cdm 5632  Rel wrel 5637  (class class class)co 7368  tpos ctpos 8177   oppFunc coppf 49481

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 5243  ax-nul 5253  ax-pr 5379

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-ne 2934  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 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-tpos 8178  df-oppf 49482

This theorem is referenced by: (None)