Theorem oppf1st2nd 49622

Description: Rewrite the opposite functor into its components (eqopi 7973). (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 6840	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐴, 𝐵⟩))
3		oppfrcl.3	. . . . . 6 ⊢ 𝐺 = ( oppFunc ‘𝐹)
4		df-ov 7365	. . . . . 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 49620	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		oppfvalg 49617	. . . . . 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 49621	. . . . 5 ⊢ (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))
13	12	iftrued 4475	. . . 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 8185	. . . . 5 ⊢ (𝐵 ∈ V → tpos 𝐵 ∈ V)
17	8, 16	simpl2im 503	. . . 4 ⊢ (𝜑 → tpos 𝐵 ∈ V)
18	15, 17	opelxpd 5665	. . 3 ⊢ (𝜑 → ⟨𝐴, tpos 𝐵⟩ ∈ (V × V))
19	14, 18	eqeltrd 2837	. 2 ⊢ (𝜑 → 𝐺 ∈ (V × V))
20	14	fveq2d 6840	. . 3 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘⟨𝐴, tpos 𝐵⟩))
21		op1stg 7949	. . . 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 6840	. . 3 ⊢ (𝜑 → (2nd ‘𝐺) = (2nd ‘⟨𝐴, tpos 𝐵⟩))
25		op2ndg 7950	. . . 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 3430  ∅c0 4274  ifcif 4467  ⟨cop 4574   × cxp 5624  dom cdm 5626  Rel wrel 5631  ‘cfv 6494  (class class class)co 7362  1^st c1st 7935  2^nd c2nd 7936  tpos ctpos 8170   oppFunc coppf 49613

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 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

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 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7937  df-2nd 7938  df-tpos 8171  df-oppf 49614

This theorem is referenced by:  2oppf  49623  funcoppc4  49635