Theorem eloppf 49294

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 2843	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ( oppFunc ‘𝐹))
4		elfvdm 6865	. . . . 5 ⊢ (𝑋 ∈ ( oppFunc ‘𝐹) → 𝐹 ∈ dom oppFunc )
5		oppffn 49285	. . . . . 6 ⊢ oppFunc Fn (V × V)
6	5	fndmi 6593	. . . . 5 ⊢ dom oppFunc = (V × V)
7	4, 6	eleqtrdi 2843	. . . 4 ⊢ (𝑋 ∈ ( oppFunc ‘𝐹) → 𝐹 ∈ (V × V))
8	3, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐹 ∈ (V × V))
9		0nelxp 5655	. . 3 ⊢ ¬ ∅ ∈ (V × V)
10		nelne2 3027	. . 3 ⊢ ((𝐹 ∈ (V × V) ∧ ¬ ∅ ∈ (V × V)) → 𝐹 ≠ ∅)
11	8, 9, 10	sylancl 586	. 2 ⊢ (𝜑 → 𝐹 ≠ ∅)
12		1st2nd2 7969	. . . . . . . 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 6835	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
15		df-ov 7358	. . . . . . 7 ⊢ ((1st ‘𝐹) oppFunc (2nd ‘𝐹)) = ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
16		fvex 6844	. . . . . . . 8 ⊢ (1st ‘𝐹) ∈ V
17		fvex 6844	. . . . . . . 8 ⊢ (2nd ‘𝐹) ∈ V
18		oppfvalg 49287	. . . . . . . 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 2758	. . . . . 6 ⊢ ( oppFunc ‘⟨(1st ‘𝐹), (2nd ‘𝐹)⟩) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅)
21	14, 20	eqtrdi 2784	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝐹) = if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅))
22	3, 21	eleqtrd 2835	. . . 4 ⊢ (𝜑 → 𝑋 ∈ if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅))
23	22	ne0d 4291	. . 3 ⊢ (𝜑 → if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅) ≠ ∅)
24		iffalse 4485	. . . 4 ⊢ (¬ (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)) → if((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩, ∅) = ∅)
25	24	necon1ai 2956	. . 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 1541   ∈ wcel 2113   ≠ wne 2929  Vcvv 3437  ∅c0 4282  ifcif 4476  ⟨cop 4583   × cxp 5619  dom cdm 5621  Rel wrel 5626  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  tpos ctpos 8164   oppFunc coppf 49283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-tpos 8165  df-oppf 49284

This theorem is referenced by:  oppc1stflem  49448