Theorem eloppf2 50060

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 2871	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐹 oppFunc 𝐺))
4		df-oppf 50049	. . . 4 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
5	4	elmpocl 7655	. . 3 ⊢ (𝑋 ∈ (𝐹 oppFunc 𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
6	3, 5	syl 18	. 2 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
7		oppfvalg 50052	. . . . . 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 2862	. . . 4 ⊢ (𝜑 → 𝑋 ∈ if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
10	9	ne0d 4288	. . 3 ⊢ (𝜑 → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) ≠ ∅)
11		iffalse 4491	. . . 4 ⊢ (¬ (Rel 𝐺 ∧ Rel dom 𝐺) → if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅) = ∅)
12	11	necon1ai 2982	. . 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 521	1 ⊢ (𝜑 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Rel 𝐺 ∧ Rel dom 𝐺)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  Vcvv 3450  ∅c0 4279  ifcif 4482  ⟨cop 4590  dom cdm 5655  Rel wrel 5660  (class class class)co 7413  tpos ctpos 8223   oppFunc coppf 50048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-tpos 8224  df-oppf 50049

This theorem is used by: (None)