Theorem oppf1st2nd 49126

Description: Rewrite the opposite functor into its components (eqopi 7960). (Contributed by Zhi Wang, 14-Nov-2025.)

Hypotheses

Ref	Expression
oppfrcl.1	⊢ (𝜑 → 𝐺 ∈ 𝑅)
oppfrcl.2	⊢ Rel 𝑅
oppfrcl.3	⊢ 𝐺 = ( oppFunc ‘𝐹)
oppfrcl2.4	⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)

Assertion

Ref	Expression
oppf1st2nd	⊢ (𝜑 → (𝐺 ∈ (V × V) ∧ ((1st ‘𝐺) = 𝐴 ∧ (2nd ‘𝐺) = tpos 𝐵)))

Proof of Theorem oppf1st2nd

Step	Hyp	Ref	Expression
1		oppfrcl2.4	. . . . . . 7 ⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)
2	1	fveq2d 6826	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐴, 𝐵⟩))
3		oppfrcl.3	. . . . . 6 ⊢ 𝐺 = ( oppFunc ‘𝐹)
4		df-ov 7352	. . . . . 6 ⊢ (𝐴 oppFunc 𝐵) = ( oppFunc ‘⟨𝐴, 𝐵⟩)
5	2, 3, 4	3eqtr4g 2789	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝐴 oppFunc 𝐵))
6		oppfrcl.1	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝑅)
7		oppfrcl.2	. . . . . . 7 ⊢ Rel 𝑅
8	6, 7, 3, 1	oppfrcl2 49124	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		oppfvalg 49121	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 oppFunc 𝐵) = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 oppFunc 𝐵) = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
11	5, 10	eqtrd 2764	. . . 4 ⊢ (𝜑 → 𝐺 = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
12	6, 7, 3, 1	oppfrcl3 49125	. . . . 5 ⊢ (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))
13	12	iftrued 4484	. . . 4 ⊢ (𝜑 → if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) = ⟨𝐴, tpos 𝐵⟩)
14	11, 13	eqtrd 2764	. . 3 ⊢ (𝜑 → 𝐺 = ⟨𝐴, tpos 𝐵⟩)
15	8	simpld 494	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
16		tposexg 8173	. . . . 5 ⊢ (𝐵 ∈ V → tpos 𝐵 ∈ V)
17	8, 16	simpl2im 503	. . . 4 ⊢ (𝜑 → tpos 𝐵 ∈ V)
18	15, 17	opelxpd 5658	. . 3 ⊢ (𝜑 → ⟨𝐴, tpos 𝐵⟩ ∈ (V × V))
19	14, 18	eqeltrd 2828	. 2 ⊢ (𝜑 → 𝐺 ∈ (V × V))
20	14	fveq2d 6826	. . 3 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘⟨𝐴, tpos 𝐵⟩))
21		op1stg 7936	. . . 4 ⊢ ((𝐴 ∈ V ∧ tpos 𝐵 ∈ V) → (1st ‘⟨𝐴, tpos 𝐵⟩) = 𝐴)
22	15, 17, 21	syl2anc 584	. . 3 ⊢ (𝜑 → (1st ‘⟨𝐴, tpos 𝐵⟩) = 𝐴)
23	20, 22	eqtrd 2764	. 2 ⊢ (𝜑 → (1st ‘𝐺) = 𝐴)
24	14	fveq2d 6826	. . 3 ⊢ (𝜑 → (2nd ‘𝐺) = (2nd ‘⟨𝐴, tpos 𝐵⟩))
25		op2ndg 7937	. . . 4 ⊢ ((𝐴 ∈ V ∧ tpos 𝐵 ∈ V) → (2nd ‘⟨𝐴, tpos 𝐵⟩) = tpos 𝐵)
26	15, 17, 25	syl2anc 584	. . 3 ⊢ (𝜑 → (2nd ‘⟨𝐴, tpos 𝐵⟩) = tpos 𝐵)
27	24, 26	eqtrd 2764	. 2 ⊢ (𝜑 → (2nd ‘𝐺) = tpos 𝐵)
28	19, 23, 27	jca32 515	1 ⊢ (𝜑 → (𝐺 ∈ (V × V) ∧ ((1st ‘𝐺) = 𝐴 ∧ (2nd ‘𝐺) = tpos 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284  ifcif 4476  ⟨cop 4583   × cxp 5617  dom cdm 5619  Rel wrel 5624  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  tpos ctpos 8158   oppFunc coppf 49117

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 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-tpos 8159  df-oppf 49118

This theorem is referenced by:  2oppf  49127  funcoppc4  49139