Theorem lrrecse 34026

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 5536	. 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 34023	. . . . 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 3402	. . 3 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} = {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))})
6		dfrab2 4241	. . . . 5 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No )
7		abid2 2881	. . . . . 6 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = (( L ‘𝑎) ∪ ( R ‘𝑎))
8	7	ineq1i 4139	. . . . 5 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
9	6, 8	eqtri 2766	. . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
10		fvex 6769	. . . . . 6 ⊢ ( L ‘𝑎) ∈ V
11		fvex 6769	. . . . . 6 ⊢ ( R ‘𝑎) ∈ V
12	10, 11	unex 7574	. . . . 5 ⊢ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∈ V
13	12	inex1 5236	. . . 4 ⊢ ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) ∈ V
14	9, 13	eqeltri 2835	. . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∈ V
15	5, 14	eqeltrdi 2847	. 2 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V)
16	1, 15	mprgbir 3078	1 ⊢ 𝑅 Se No

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {cab 2715  {crab 3067  Vcvv 3422   ∪ cun 3881   ∩ cin 3882   class class class wbr 5070  {copab 5132   Se wse 5533  ‘cfv 6418   No csur 33770   L cleft 33956   R cright 33957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-se 5536  df-iota 6376  df-fv 6426

This theorem is referenced by:  noinds  34029  norecfn  34030  norecov  34031  noxpordse  34036  no2indslem  34038  no3inds  34042