Theorem oppf1st2nd 49757

Description: Rewrite the opposite functor into its components (eqopi 8008). (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 6873	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐴, 𝐵⟩))
3		oppfrcl.3	. . . . . 6 ⊢ 𝐺 = ( oppFunc ‘𝐹)
4		df-ov 7401	. . . . . 6 ⊢ (𝐴 oppFunc 𝐵) = ( oppFunc ‘⟨𝐴, 𝐵⟩)
5	2, 3, 4	3eqtr4g 2824	. . . . 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 49755	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		oppfvalg 49752	. . . . . 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 2799	. . . 4 ⊢ (𝜑 → 𝐺 = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
12	6, 7, 3, 1	oppfrcl3 49756	. . . . 5 ⊢ (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))
13	12	iftrued 4490	. . . 4 ⊢ (𝜑 → if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) = ⟨𝐴, tpos 𝐵⟩)
14	11, 13	eqtrd 2799	. . 3 ⊢ (𝜑 → 𝐺 = ⟨𝐴, tpos 𝐵⟩)
15	8	simpld 498	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
16		tposexg 8222	. . . . 5 ⊢ (𝐵 ∈ V → tpos 𝐵 ∈ V)
17	8, 16	simpl2im 511	. . . 4 ⊢ (𝜑 → tpos 𝐵 ∈ V)
18	15, 17	opelxpd 5688	. . 3 ⊢ (𝜑 → ⟨𝐴, tpos 𝐵⟩ ∈ (V × V))
19	14, 18	eqeltrd 2864	. 2 ⊢ (𝜑 → 𝐺 ∈ (V × V))
20	14	fveq2d 6873	. . 3 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘⟨𝐴, tpos 𝐵⟩))
21		op1stg 7984	. . . 4 ⊢ ((𝐴 ∈ V ∧ tpos 𝐵 ∈ V) → (1st ‘⟨𝐴, tpos 𝐵⟩) = 𝐴)
22	15, 17, 21	syl2anc 593	. . 3 ⊢ (𝜑 → (1st ‘⟨𝐴, tpos 𝐵⟩) = 𝐴)
23	20, 22	eqtrd 2799	. 2 ⊢ (𝜑 → (1st ‘𝐺) = 𝐴)
24	14	fveq2d 6873	. . 3 ⊢ (𝜑 → (2nd ‘𝐺) = (2nd ‘⟨𝐴, tpos 𝐵⟩))
25		op2ndg 7985	. . . 4 ⊢ ((𝐴 ∈ V ∧ tpos 𝐵 ∈ V) → (2nd ‘⟨𝐴, tpos 𝐵⟩) = tpos 𝐵)
26	15, 17, 25	syl2anc 593	. . 3 ⊢ (𝜑 → (2nd ‘⟨𝐴, tpos 𝐵⟩) = tpos 𝐵)
27	24, 26	eqtrd 2799	. 2 ⊢ (𝜑 → (2nd ‘𝐺) = tpos 𝐵)
28	19, 23, 27	jca32 523	1 ⊢ (𝜑 → (𝐺 ∈ (V × V) ∧ ((1st ‘𝐺) = 𝐴 ∧ (2nd ‘𝐺) = tpos 𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456  ∅c0 4287  ifcif 4482  ⟨cop 4590   × cxp 5647  dom cdm 5649  Rel wrel 5654  ‘cfv 6523  (class class class)co 7398  1^st c1st 7970  2^nd c2nd 7971  tpos ctpos 8207   oppFunc coppf 49748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-tpos 8208  df-oppf 49749

This theorem is referenced by:  2oppf  49758  funcoppc4  49770