Theorem lrrecse 27888

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 5575	. 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 27885	. . . . 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 4269	. . . . 5 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No )
7		abid2 2870	. . . . . 6 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = (( L ‘𝑎) ∪ ( R ‘𝑎))
8	7	ineq1i 4165	. . . . 5 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
9	6, 8	eqtri 2756	. . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
10		fvex 6843	. . . . . 6 ⊢ ( L ‘𝑎) ∈ V
11		fvex 6843	. . . . . 6 ⊢ ( R ‘𝑎) ∈ V
12	10, 11	unex 7685	. . . . 5 ⊢ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∈ V
13	12	inex1 5259	. . . 4 ⊢ ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) ∈ V
14	9, 13	eqeltri 2829	. . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∈ V
15	5, 14	eqeltrdi 2841	. 2 ⊢ (𝑎 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝑎} ∈ V)
16	1, 15	mprgbir 3055	1 ⊢ 𝑅 Se No

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {cab 2711  {crab 3396  Vcvv 3437   ∪ cun 3896   ∩ cin 3897   class class class wbr 5095  {copab 5157   Se wse 5572  ‘cfv 6488   No csur 27581   L cleft 27789   R cright 27790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-se 5575  df-iota 6444  df-fv 6496

This theorem is referenced by:  noinds  27891  norecfn  27892  norecov  27893  noxpordse  27898  no2indslem  27900  no3inds  27904