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Theorem lrrecse 33742
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 5484 . 2 (𝑅 Se No ↔ ∀𝑎 No {𝑏 No 𝑏𝑅𝑎} ∈ V)
2 lrrec.1 . . . . . 6 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
32lrrecval 33739 . . . . 5 ((𝑏 No 𝑎 No ) → (𝑏𝑅𝑎𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))))
43ancoms 462 . . . 4 ((𝑎 No 𝑏 No ) → (𝑏𝑅𝑎𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))))
54rabbidva 3379 . . 3 (𝑎 No → {𝑏 No 𝑏𝑅𝑎} = {𝑏 No 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))})
6 dfrab2 4199 . . . . 5 {𝑏 No 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ({𝑏𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No )
7 abid2 2874 . . . . . 6 {𝑏𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = (( L ‘𝑎) ∪ ( R ‘𝑎))
87ineq1i 4099 . . . . 5 ({𝑏𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∩ No ) = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
96, 8eqtri 2761 . . . 4 {𝑏 No 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} = ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No )
10 fvex 6687 . . . . . 6 ( L ‘𝑎) ∈ V
11 fvex 6687 . . . . . 6 ( R ‘𝑎) ∈ V
1210, 11unex 7487 . . . . 5 (( L ‘𝑎) ∪ ( R ‘𝑎)) ∈ V
1312inex1 5185 . . . 4 ((( L ‘𝑎) ∪ ( R ‘𝑎)) ∩ No ) ∈ V
149, 13eqeltri 2829 . . 3 {𝑏 No 𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))} ∈ V
155, 14eqeltrdi 2841 . 2 (𝑎 No → {𝑏 No 𝑏𝑅𝑎} ∈ V)
161, 15mprgbir 3068 1 𝑅 Se No
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1542  wcel 2114  {cab 2716  {crab 3057  Vcvv 3398  cun 3841  cin 3842   class class class wbr 5030  {copab 5092   Se wse 5481  cfv 6339   No csur 33486   L cleft 33672   R cright 33673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-se 5484  df-iota 6297  df-fv 6347
This theorem is referenced by:  noinds  33745  norecfn  33746  norecov  33747  noxpordse  33752  no2indslem  33754  no3inds  33758
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