Theorem oppfrcl3 49518

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 6848	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨𝐴, 𝐵⟩))
3		oppfrcl.3	. . . . 5 ⊢ 𝐺 = ( oppFunc ‘𝐹)
4		df-ov 7373	. . . . 5 ⊢ (𝐴 oppFunc 𝐵) = ( oppFunc ‘⟨𝐴, 𝐵⟩)
5	2, 3, 4	3eqtr4g 2797	. . . 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 49517	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		oppfvalg 49514	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 oppFunc 𝐵) = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 oppFunc 𝐵) = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
11	5, 10	eqtrd 2772	. . 3 ⊢ (𝜑 → 𝐺 = if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅))
12	6, 7	oppfrcllem 49515	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
13	11, 12	eqnetrrd 3001	. 2 ⊢ (𝜑 → if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) ≠ ∅)
14		iffalse 4490	. . 3 ⊢ (¬ (Rel 𝐵 ∧ Rel dom 𝐵) → if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) = ∅)
15	14	necon1ai 2960	. 2 ⊢ (if((Rel 𝐵 ∧ Rel dom 𝐵), ⟨𝐴, tpos 𝐵⟩, ∅) ≠ ∅ → (Rel 𝐵 ∧ Rel dom 𝐵))
16	13, 15	syl 17	1 ⊢ (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442  ∅c0 4287  ifcif 4481  ⟨cop 4588  dom cdm 5634  Rel wrel 5639  ‘cfv 6502  (class class class)co 7370  tpos ctpos 8179   oppFunc coppf 49510

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 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

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 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-tpos 8180  df-oppf 49511

This theorem is referenced by:  oppf1st2nd  49519  2oppf  49520  funcoppc4  49532