Theorem oppf1st2nd 49318

Description: Rewrite the opposite functor into its components (eqopi 7967). (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 6836	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐴, 𝐵⟩))
3		oppfrcl.3	. . . . . 6 ⊢ 𝐺 = ( oppFunc ‘𝐹)
4		df-ov 7359	. . . . . 6 ⊢ (𝐴 oppFunc 𝐵) = ( oppFunc ‘⟨𝐴, 𝐵⟩)
5	2, 3, 4	3eqtr4g 2794	. . . . 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 49316	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		oppfvalg 49313	. . . . . 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 2769	. . . 4 ⊢ (𝜑 → 𝐺 = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
12	6, 7, 3, 1	oppfrcl3 49317	. . . . 5 ⊢ (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))
13	12	iftrued 4485	. . . 4 ⊢ (𝜑 → if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) = ⟨𝐴, tpos 𝐵⟩)
14	11, 13	eqtrd 2769	. . 3 ⊢ (𝜑 → 𝐺 = ⟨𝐴, tpos 𝐵⟩)
15	8	simpld 494	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
16		tposexg 8180	. . . . 5 ⊢ (𝐵 ∈ V → tpos 𝐵 ∈ V)
17	8, 16	simpl2im 503	. . . 4 ⊢ (𝜑 → tpos 𝐵 ∈ V)
18	15, 17	opelxpd 5661	. . 3 ⊢ (𝜑 → ⟨𝐴, tpos 𝐵⟩ ∈ (V × V))
19	14, 18	eqeltrd 2834	. 2 ⊢ (𝜑 → 𝐺 ∈ (V × V))
20	14	fveq2d 6836	. . 3 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘⟨𝐴, tpos 𝐵⟩))
21		op1stg 7943	. . . 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 2769	. 2 ⊢ (𝜑 → (1st ‘𝐺) = 𝐴)
24	14	fveq2d 6836	. . 3 ⊢ (𝜑 → (2nd ‘𝐺) = (2nd ‘⟨𝐴, tpos 𝐵⟩))
25		op2ndg 7944	. . . 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 2769	. 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 1541   ∈ wcel 2113  Vcvv 3438  ∅c0 4283  ifcif 4477  ⟨cop 4584   × cxp 5620  dom cdm 5622  Rel wrel 5627  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  tpos ctpos 8165   oppFunc coppf 49309

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 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-tpos 8166  df-oppf 49310

This theorem is referenced by:  2oppf  49319  funcoppc4  49331