Theorem lrrecse 28361

Description: Next, we show that 𝑅 is set-like over No. (Contributed by Scott Fenton, 19-Aug-2024.)

Hypothesis

Ref	Expression
lrrec.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}

Assertion

Ref	Expression
lrrecse	⊢ 𝑅 Se No

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝑅(𝑥, 𝑦)

Proof of Theorem lrrecse

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-se 5610	. 2 ⊢ (𝑅 Se No ↔ ∀𝑎 ∈ No {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V)
2		lrrec.1	. . . . . 6 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}
3	2	lrrecval 28358	. . . . 5 ⊢ ((𝑏 ∈ No ∧ 𝑎 ∈ No) → (𝑏𝑅𝑎 ↔ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))))
4	3	ancoms 464	. . . 4 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No) → (𝑏𝑅𝑎 ↔ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))))
5	4	rabbidva 3422	. . 3 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} = {𝑏 ∈ No ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))})
6		dfrab2 4268	. . . . 5 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} = ({𝑏 ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} ∩ No)
7		abid2 2901	. . . . . 6 ⊢ {𝑏 ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} = ((L‘𝑎) ∪ (R‘𝑎))
8	7	ineq1i 4116	. . . . 5 ⊢ ({𝑏 ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} ∩ No) = (((L‘𝑎) ∪ (R‘𝑎)) ∩ No)
9	6, 8	eqtri 2786	. . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} = (((L‘𝑎) ∪ (R‘𝑎)) ∩ No)
10		fvex 6924	. . . . . 6 ⊢ (L‘𝑎) ∈ V
11		fvex 6924	. . . . . 6 ⊢ (R‘𝑎) ∈ V
12	10, 11	unex 7788	. . . . 5 ⊢ ((L‘𝑎) ∪ (R‘𝑎)) ∈ V
13	12	inex1 5282	. . . 4 ⊢ (((L‘𝑎) ∪ (R‘𝑎)) ∩ No) ∈ V
14	9, 13	eqeltri 2860	. . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} ∈ V
15	5, 14	eqeltrdi 2872	. 2 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V)
16	1, 15	mprgbir 3087	1 ⊢ 𝑅 Se No

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {cab 2741  {crab 3416  Vcvv 3454   ∩ cin 4092   ∪ cun 4152   class class class wbr 5106  {copab 5171   Se wse 5607  ‘cfv 6561  Nocsur 28030  Lcleft 28244  Rcright 28245

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-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

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-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  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-br 5107  df-opab 5172  df-se 5610  df-iota 6515  df-fv 6569

This theorem is used by:  noinds  28364  norecfn  28365  norecov  28366  noxpordse  28371  no2indlesm  28373  no3inds  28377