Theorem eloppf 49492

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 2847	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ( oppFunc ‘𝐹))
4		elfvdm 6876	. . . . 5 ⊢ (𝑋 ∈ ( oppFunc ‘𝐹) → 𝐹 ∈ dom oppFunc )
5		oppffn 49483	. . . . . 6 ⊢ oppFunc Fn (V × V)
6	5	fndmi 6604	. . . . 5 ⊢ dom oppFunc = (V × V)
7	4, 6	eleqtrdi 2847	. . . 4 ⊢ (𝑋 ∈ ( oppFunc ‘𝐹) → 𝐹 ∈ (V × V))
8	3, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐹 ∈ (V × V))
9		0nelxp 5666	. . 3 ⊢ ¬ ∅ ∈ (V × V)
10		nelne2 3031	. . 3 ⊢ ((𝐹 ∈ (V × V) ∧ ¬ ∅ ∈ (V × V)) → 𝐹 ≠ ∅)
11	8, 9, 10	sylancl 587	. 2 ⊢ (𝜑 → 𝐹 ≠ ∅)
12		1st2nd2 7982	. . . . . . . 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 6846	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
15		df-ov 7371	. . . . . . 7 ⊢ ((1st ‘𝐹) oppFunc (2nd ‘𝐹)) = ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
16		fvex 6855	. . . . . . . 8 ⊢ (1st ‘𝐹) ∈ V
17		fvex 6855	. . . . . . . 8 ⊢ (2nd ‘𝐹) ∈ V
18		oppfvalg 49485	. . . . . . . 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 693	. . . . . . 7 ⊢ ((1st ‘𝐹) oppFunc (2nd ‘𝐹)) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅)
20	15, 19	eqtr3i 2762	. . . . . 6 ⊢ ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅)
21	14, 20	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝐹) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅))
22	3, 21	eleqtrd 2839	. . . 4 ⊢ (𝜑 → 𝑋 ∈ if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅))
23	22	ne0d 4296	. . 3 ⊢ (𝜑 → if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅) ≠ ∅)
24		iffalse 4490	. . . 4 ⊢ (¬ (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)) → if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅) = ∅)
25	24	necon1ai 2960	. . 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 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442  ∅c0 4287  ifcif 4481  ⟨cop 4588   × cxp 5630  dom cdm 5632  Rel wrel 5637  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  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  ax-un 7690

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-csb 3852  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-iun 4950  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-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-tpos 8178  df-oppf 49482

This theorem is referenced by:  oppc1stflem  49646