Theorem lrrecse 27855

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 5594	. 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 27852	. . . . 5 ⊢ ((𝑏 ∈ No ∧ 𝑎 ∈ No ) → (𝑏𝑅𝑎 ↔ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))))
4	3	ancoms 458	. . . 4 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ) → (𝑏𝑅𝑎 ↔ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))))
5	4	rabbidva 3415	. . 3 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} = {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))})
6		dfrab2 4285	. . . . 5 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No )
7		abid2 2866	. . . . . 6 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = (( L ‘𝑎) ∪ ( R ‘𝑎))
8	7	ineq1i 4181	. . . . 5 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
9	6, 8	eqtri 2753	. . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
10		fvex 6873	. . . . . 6 ⊢ ( L ‘𝑎) ∈ V
11		fvex 6873	. . . . . 6 ⊢ ( R ‘𝑎) ∈ V
12	10, 11	unex 7722	. . . . 5 ⊢ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∈ V
13	12	inex1 5274	. . . 4 ⊢ ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) ∈ V
14	9, 13	eqeltri 2825	. . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∈ V
15	5, 14	eqeltrdi 2837	. 2 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V)
16	1, 15	mprgbir 3052	1 ⊢ 𝑅 Se No

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2708  {crab 3408  Vcvv 3450   ∪ cun 3914   ∩ cin 3915   class class class wbr 5109  {copab 5171   Se wse 5591  ‘cfv 6513   No csur 27557   L cleft 27759   R cright 27760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-se 5594  df-iota 6466  df-fv 6521

This theorem is referenced by:  noinds  27858  norecfn  27859  norecov  27860  noxpordse  27865  no2indslem  27867  no3inds  27871