Theorem oppf1st2nd 49443

Description: Rewrite the opposite functor into its components (eqopi 7971). (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 6839	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐴, 𝐵⟩))
3		oppfrcl.3	. . . . . 6 ⊢ 𝐺 = ( oppFunc ‘𝐹)
4		df-ov 7363	. . . . . 6 ⊢ (𝐴 oppFunc 𝐵) = ( oppFunc ‘⟨𝐴, 𝐵⟩)
5	2, 3, 4	3eqtr4g 2797	. . . . 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 49441	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		oppfvalg 49438	. . . . . 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 2772	. . . 4 ⊢ (𝜑 → 𝐺 = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
12	6, 7, 3, 1	oppfrcl3 49442	. . . . 5 ⊢ (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))
13	12	iftrued 4488	. . . 4 ⊢ (𝜑 → if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) = ⟨𝐴, tpos 𝐵⟩)
14	11, 13	eqtrd 2772	. . 3 ⊢ (𝜑 → 𝐺 = ⟨𝐴, tpos 𝐵⟩)
15	8	simpld 494	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
16		tposexg 8184	. . . . 5 ⊢ (𝐵 ∈ V → tpos 𝐵 ∈ V)
17	8, 16	simpl2im 503	. . . 4 ⊢ (𝜑 → tpos 𝐵 ∈ V)
18	15, 17	opelxpd 5664	. . 3 ⊢ (𝜑 → ⟨𝐴, tpos 𝐵⟩ ∈ (V × V))
19	14, 18	eqeltrd 2837	. 2 ⊢ (𝜑 → 𝐺 ∈ (V × V))
20	14	fveq2d 6839	. . 3 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘⟨𝐴, tpos 𝐵⟩))
21		op1stg 7947	. . . 4 ⊢ ((𝐴 ∈ V ∧ tpos 𝐵 ∈ V) → (1st ‘⟨𝐴, tpos 𝐵⟩) = 𝐴)
22	15, 17, 21	syl2anc 585	. . 3 ⊢ (𝜑 → (1st ‘⟨𝐴, tpos 𝐵⟩) = 𝐴)
23	20, 22	eqtrd 2772	. 2 ⊢ (𝜑 → (1st ‘𝐺) = 𝐴)
24	14	fveq2d 6839	. . 3 ⊢ (𝜑 → (2nd ‘𝐺) = (2nd ‘⟨𝐴, tpos 𝐵⟩))
25		op2ndg 7948	. . . 4 ⊢ ((𝐴 ∈ V ∧ tpos 𝐵 ∈ V) → (2nd ‘⟨𝐴, tpos 𝐵⟩) = tpos 𝐵)
26	15, 17, 25	syl2anc 585	. . 3 ⊢ (𝜑 → (2nd ‘⟨𝐴, tpos 𝐵⟩) = tpos 𝐵)
27	24, 26	eqtrd 2772	. 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 1542   ∈ wcel 2114  Vcvv 3441  ∅c0 4286  ifcif 4480  ⟨cop 4587   × cxp 5623  dom cdm 5625  Rel wrel 5630  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  tpos ctpos 8169   oppFunc coppf 49434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-tpos 8170  df-oppf 49435

This theorem is referenced by:  2oppf  49444  funcoppc4  49456