Theorem eloppf 49791

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 2879	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ( oppFunc ‘𝐹))
4		elfvdm 6913	. . . . 5 ⊢ (𝑋 ∈ ( oppFunc ‘𝐹) → 𝐹 ∈ dom oppFunc )
5		oppffn 49782	. . . . . 6 ⊢ oppFunc Fn (V × V)
6	5	fndmi 6637	. . . . 5 ⊢ dom oppFunc = (V × V)
7	4, 6	eleqtrdi 2879	. . . 4 ⊢ (𝑋 ∈ ( oppFunc ‘𝐹) → 𝐹 ∈ (V × V))
8	3, 7	syl 18	. . 3 ⊢ (𝜑 → 𝐹 ∈ (V × V))
9		0nelxp 5693	. . 3 ⊢ ¬ ∅ ∈ (V × V)
10		nelne2 3062	. . 3 ⊢ ((𝐹 ∈ (V × V) ∧ ¬ ∅ ∈ (V × V)) → 𝐹 ≠ ∅)
11	8, 9, 10	sylancl 597	. 2 ⊢ (𝜑 → 𝐹 ≠ ∅)
12		1st2nd2 8021	. . . . . . . 8 ⊢ (𝐹 ∈ (V × V) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
13	3, 7, 12	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
14	13	fveq2d 6883	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
15		df-ov 7411	. . . . . . 7 ⊢ ((1st ‘𝐹) oppFunc (2nd ‘𝐹)) = ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
16		fvex 6892	. . . . . . . 8 ⊢ (1st ‘𝐹) ∈ V
17		fvex 6892	. . . . . . . 8 ⊢ (2nd ‘𝐹) ∈ V
18		oppfvalg 49784	. . . . . . . 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 704	. . . . . . 7 ⊢ ((1st ‘𝐹) oppFunc (2nd ‘𝐹)) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅)
20	15, 19	eqtr3i 2794	. . . . . 6 ⊢ ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅)
21	14, 20	eqtrdi 2820	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝐹) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅))
22	3, 21	eleqtrd 2871	. . . 4 ⊢ (𝜑 → 𝑋 ∈ if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅))
23	22	ne0d 4303	. . 3 ⊢ (𝜑 → if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅) ≠ ∅)
24		iffalse 4498	. . . 4 ⊢ (¬ (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)) → if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅) = ∅)
25	24	necon1ai 2991	. . 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 18	. 2 ⊢ (𝜑 → (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)))
27	11, 26	jca 520	1 ⊢ (𝜑 → (𝐹 ≠ ∅ ∧ (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  ∅c0 4294  ifcif 4489  ⟨cop 4597   × cxp 5657  dom cdm 5659  Rel wrel 5664  ‘cfv 6534  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  tpos ctpos 8217   oppFunc coppf 49780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-tpos 8218  df-oppf 49781

This theorem is referenced by:  oppc1stflem  49945