Theorem oppfrcl3 50449

Description: If an opposite functor of a class is a functor, then the second component of the original class must be a relation whose domain is a relation as well. (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
oppfrcl3	⊢ (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))

Proof of Theorem oppfrcl3

Step	Hyp	Ref	Expression
1		oppfrcl2.4	. . . . . 6 ⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)
2	1	fveq2d 6909	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐴, 𝐵⟩))
3		oppfrcl.3	. . . . 5 ⊢ 𝐺 = ( oppFunc ‘𝐹)
4		df-ov 7443	. . . . 5 ⊢ (𝐴 oppFunc 𝐵) = ( oppFunc ‘⟨𝐴, 𝐵⟩)
5	2, 3, 4	3eqtr4g 2823	. . . 4 ⊢ (𝜑 → 𝐺 = (𝐴 oppFunc 𝐵))
6		oppfrcl.1	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑅)
7		oppfrcl.2	. . . . . 6 ⊢ Rel 𝑅
8	6, 7, 3, 1	oppfrcl2 50448	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		oppfvalg 50445	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 oppFunc 𝐵) = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → (𝐴 oppFunc 𝐵) = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
11	5, 10	eqtrd 2798	. . 3 ⊢ (𝜑 → 𝐺 = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
12	6, 7	oppfrcllem 50446	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
13	11, 12	eqnetrrd 3027	. 2 ⊢ (𝜑 → if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) ≠ ∅)
14		iffalse 4494	. . 3 ⊢ (¬ (Rel 𝐵 ∧ Rel dom 𝐵) → if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) = ∅)
15	14	necon1ai 2986	. 2 ⊢ (if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) ≠ ∅ → (Rel 𝐵 ∧ Rel dom 𝐵))
16	13, 15	syl 18	1 ⊢ (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  ifcif 4485  ⟨cop 4838  dom cdm 5655  Rel wrel 5660  ‘cfv 6555  (class class class)co 7440  tpos ctpos 8267   oppFunc coppf 50441

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-tpos 8268  df-oppf 50442

This theorem is used by:  oppf1st2nd  50450  2oppf  50451  funcoppc4  50463