Theorem lrrecse 28334

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 5604	. 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 28331	. . . . 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 3418	. . 3 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} = {𝑏 ∈ No ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))})
6		dfrab2 4265	. . . . 5 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} = ({𝑏 ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} ∩ No)
7		abid2 2897	. . . . . 6 ⊢ {𝑏 ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} = ((L‘𝑎) ∪ (R‘𝑎))
8	7	ineq1i 4113	. . . . 5 ⊢ ({𝑏 ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} ∩ No) = (((L‘𝑎) ∪ (R‘𝑎)) ∩ No)
9	6, 8	eqtri 2783	. . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} = (((L‘𝑎) ∪ (R‘𝑎)) ∩ No)
10		fvex 6899	. . . . . 6 ⊢ (L‘𝑎) ∈ V
11		fvex 6899	. . . . . 6 ⊢ (R‘𝑎) ∈ V
12	10, 11	unex 7762	. . . . 5 ⊢ ((L‘𝑎) ∪ (R‘𝑎)) ∈ V
13	12	inex1 5276	. . . 4 ⊢ (((L‘𝑎) ∪ (R‘𝑎)) ∩ No) ∈ V
14	9, 13	eqeltri 2856	. . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))} ∈ V
15	5, 14	eqeltrdi 2868	. 2 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V)
16	1, 15	mprgbir 3083	1 ⊢ 𝑅 Se No

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  {cab 2738  {crab 3412  Vcvv 3450   ∩ cin 4089   ∪ cun 4149   class class class wbr 5102  {copab 5166   Se wse 5601  ‘cfv 6539  Nocsur 28003  Lcleft 28217  Rcright 28218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-se 5604  df-iota 6494  df-fv 6547

This theorem is used by:  noinds  28337  norecfn  28338  norecov  28339  noxpordse  28344  no2indlesm  28346  no3inds  28350