Theorem eloppf 49095

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

Hypotheses

Ref	Expression
eloppf.g	⊢ 𝐺 = ( oppFunc ‘𝐹)
eloppf.x	⊢ (𝜑 → 𝑋 ∈ 𝐺)

Assertion

Ref	Expression
eloppf	⊢ (𝜑 → (𝐹 ≠ ∅ ∧ (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹))))

Proof of Theorem eloppf

Step	Hyp	Ref	Expression
1		eloppf.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐺)
2		eloppf.g	. . . . 5 ⊢ 𝐺 = ( oppFunc ‘𝐹)
3	1, 2	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ( oppFunc ‘𝐹))
4		elfvdm 6877	. . . . 5 ⊢ (𝑋 ∈ ( oppFunc ‘𝐹) → 𝐹 ∈ dom oppFunc )
5		oppffn 49086	. . . . . 6 ⊢ oppFunc Fn (V × V)
6	5	fndmi 6604	. . . . 5 ⊢ dom oppFunc = (V × V)
7	4, 6	eleqtrdi 2838	. . . 4 ⊢ (𝑋 ∈ ( oppFunc ‘𝐹) → 𝐹 ∈ (V × V))
8	3, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐹 ∈ (V × V))
9		0nelxp 5665	. . 3 ⊢ ¬ ∅ ∈ (V × V)
10		nelne2 3023	. . 3 ⊢ ((𝐹 ∈ (V × V) ∧ ¬ ∅ ∈ (V × V)) → 𝐹 ≠ ∅)
11	8, 9, 10	sylancl 586	. 2 ⊢ (𝜑 → 𝐹 ≠ ∅)
12		1st2nd2 7986	. . . . . . . 8 ⊢ (𝐹 ∈ (V × V) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
13	3, 7, 12	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
14	13	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
15		df-ov 7372	. . . . . . 7 ⊢ ((1st ‘𝐹) oppFunc (2nd ‘𝐹)) = ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
16		fvex 6853	. . . . . . . 8 ⊢ (1st ‘𝐹) ∈ V
17		fvex 6853	. . . . . . . 8 ⊢ (2nd ‘𝐹) ∈ V
18		oppfvalg 49088	. . . . . . . 8 ⊢ (((1st ‘𝐹) ∈ V ∧ (2nd ‘𝐹) ∈ V) → ((1st ‘𝐹) oppFunc (2nd ‘𝐹)) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅))
19	16, 17, 18	mp2an 692	. . . . . . 7 ⊢ ((1st ‘𝐹) oppFunc (2nd ‘𝐹)) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅)
20	15, 19	eqtr3i 2754	. . . . . 6 ⊢ ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅)
21	14, 20	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝐹) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅))
22	3, 21	eleqtrd 2830	. . . 4 ⊢ (𝜑 → 𝑋 ∈ if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅))
23	22	ne0d 4301	. . 3 ⊢ (𝜑 → if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅) ≠ ∅)
24		iffalse 4493	. . . 4 ⊢ (¬ (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)) → if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅) = ∅)
25	24	necon1ai 2952	. . 3 ⊢ (if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅) ≠ ∅ → (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)))
26	23, 25	syl 17	. 2 ⊢ (𝜑 → (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)))
27	11, 26	jca 511	1 ⊢ (𝜑 → (𝐹 ≠ ∅ ∧ (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3444  ∅c0 4292  ifcif 4484  ⟨cop 4591   × cxp 5629  dom cdm 5631  Rel wrel 5636  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  tpos ctpos 8181   oppFunc coppf 49084

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 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-tpos 8182  df-oppf 49085

This theorem is referenced by:  oppc1stflem  49249