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Theorem noxpordse 27267
Description: Next we establish the set-like nature of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
Hypotheses
Ref Expression
noxpord.1 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
noxpord.2 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st𝑥)𝑅(1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥)𝑅(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}
Assertion
Ref Expression
noxpordse 𝑆 Se ( No × No )
Distinct variable groups:   𝑥,𝑅,𝑦   𝑎,𝑏
Allowed substitution hints:   𝑅(𝑎,𝑏)   𝑆(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem noxpordse
StepHypRef Expression
1 noxpord.2 . . 3 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st𝑥)𝑅(1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥)𝑅(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}
2 noxpord.1 . . . . 5 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
32lrrecse 27257 . . . 4 𝑅 Se No
43a1i 11 . . 3 (⊤ → 𝑅 Se No )
51, 4, 4sexp2 8079 . 2 (⊤ → 𝑆 Se ( No × No ))
65mptru 1549 1 𝑆 Se ( No × No )
Colors of variables: wff setvar class
Syntax hints:  wo 846  w3a 1088   = wceq 1542  wtru 1543  wcel 2107  wne 2944  cun 3909   class class class wbr 5106  {copab 5168   Se wse 5587   × cxp 5632  cfv 6497  1st c1st 7920  2nd c2nd 7921   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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  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-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  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-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-se 5590  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  norec2fn  27271  norec2ov  27272
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