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Theorem lrrecse 27257
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.
StepHypRef Expression
1 df-se 5590 . 2 (𝑅 Se No ↔ ∀𝑎 No {𝑏 No 𝑏𝑅𝑎} ∈ V)
2 lrrec.1 . . . . . 6 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
32lrrecval 27254 . . . . 5 ((𝑏 No 𝑎 No ) → (𝑏𝑅𝑎𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))))
43ancoms 460 . . . 4 ((𝑎 No 𝑏 No ) → (𝑏𝑅𝑎𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))))
54rabbidva 3415 . . 3 (𝑎 No → {𝑏 No 𝑏𝑅𝑎} = {𝑏 No 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))})
6 dfrab2 4271 . . . . 5 {𝑏 No 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ({𝑏𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No )
7 abid2 2876 . . . . . 6 {𝑏𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = (( L ‘𝑎) ∪ ( R ‘𝑎))
87ineq1i 4169 . . . . 5 ({𝑏𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
96, 8eqtri 2765 . . . 4 {𝑏 No 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
10 fvex 6856 . . . . . 6 ( L ‘𝑎) ∈ V
11 fvex 6856 . . . . . 6 ( R ‘𝑎) ∈ V
1210, 11unex 7681 . . . . 5 (( L ‘𝑎) ∪ ( R ‘𝑎)) ∈ V
1312inex1 5275 . . . 4 ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) ∈ V
149, 13eqeltri 2834 . . 3 {𝑏 No 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∈ V
155, 14eqeltrdi 2846 . 2 (𝑎 No → {𝑏 No 𝑏𝑅𝑎} ∈ V)
161, 15mprgbir 3072 1 𝑅 Se No
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  {cab 2714  {crab 3408  Vcvv 3446  cun 3909  cin 3910   class class class wbr 5106  {copab 5168   Se wse 5587  cfv 6497   No csur 26991   L cleft 27178   R cright 27179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-se 5590  df-iota 6449  df-fv 6505
This theorem is referenced by:  noinds  27260  norecfn  27261  norecov  27262  noxpordse  27267  no2indslem  27269  no3inds  27273
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