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Theorem lrrecfr 27258
Description: Now we show that 𝑅 is founded 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
lrrecfr 𝑅 Fr No
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem lrrecfr
Dummy variables 𝑎 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fr 5589 . 2 (𝑅 Fr No ↔ ∀𝑎((𝑎 No 𝑎 ≠ ∅) → ∃𝑝𝑎𝑞𝑎 ¬ 𝑞𝑅𝑝))
2 bdayfun 27115 . . . . 5 Fun bday
3 imassrn 6025 . . . . . . 7 ( bday 𝑎) ⊆ ran bday
4 bdayrn 27118 . . . . . . 7 ran bday = On
53, 4sseqtri 3981 . . . . . 6 ( bday 𝑎) ⊆ On
6 fvex 6856 . . . . . . . . . . . . 13 ( bday 𝑞) ∈ V
76jctr 526 . . . . . . . . . . . 12 (𝑞𝑎 → (𝑞𝑎 ∧ ( bday 𝑞) ∈ V))
87eximi 1838 . . . . . . . . . . 11 (∃𝑞 𝑞𝑎 → ∃𝑞(𝑞𝑎 ∧ ( bday 𝑞) ∈ V))
9 n0 4307 . . . . . . . . . . 11 (𝑎 ≠ ∅ ↔ ∃𝑞 𝑞𝑎)
10 df-rex 3075 . . . . . . . . . . 11 (∃𝑞𝑎 ( bday 𝑞) ∈ V ↔ ∃𝑞(𝑞𝑎 ∧ ( bday 𝑞) ∈ V))
118, 9, 103imtr4i 292 . . . . . . . . . 10 (𝑎 ≠ ∅ → ∃𝑞𝑎 ( bday 𝑞) ∈ V)
12 isset 3459 . . . . . . . . . . . . 13 (( bday 𝑞) ∈ V ↔ ∃𝑝 𝑝 = ( bday 𝑞))
13 eqcom 2744 . . . . . . . . . . . . . 14 (𝑝 = ( bday 𝑞) ↔ ( bday 𝑞) = 𝑝)
1413exbii 1851 . . . . . . . . . . . . 13 (∃𝑝 𝑝 = ( bday 𝑞) ↔ ∃𝑝( bday 𝑞) = 𝑝)
1512, 14bitri 275 . . . . . . . . . . . 12 (( bday 𝑞) ∈ V ↔ ∃𝑝( bday 𝑞) = 𝑝)
1615rexbii 3098 . . . . . . . . . . 11 (∃𝑞𝑎 ( bday 𝑞) ∈ V ↔ ∃𝑞𝑎𝑝( bday 𝑞) = 𝑝)
17 rexcom4 3272 . . . . . . . . . . 11 (∃𝑞𝑎𝑝( bday 𝑞) = 𝑝 ↔ ∃𝑝𝑞𝑎 ( bday 𝑞) = 𝑝)
1816, 17bitri 275 . . . . . . . . . 10 (∃𝑞𝑎 ( bday 𝑞) ∈ V ↔ ∃𝑝𝑞𝑎 ( bday 𝑞) = 𝑝)
1911, 18sylib 217 . . . . . . . . 9 (𝑎 ≠ ∅ → ∃𝑝𝑞𝑎 ( bday 𝑞) = 𝑝)
2019adantl 483 . . . . . . . 8 ((𝑎 No 𝑎 ≠ ∅) → ∃𝑝𝑞𝑎 ( bday 𝑞) = 𝑝)
21 bdayfn 27116 . . . . . . . . . . 11 bday Fn No
22 fvelimab 6915 . . . . . . . . . . 11 (( bday Fn No 𝑎 No ) → (𝑝 ∈ ( bday 𝑎) ↔ ∃𝑞𝑎 ( bday 𝑞) = 𝑝))
2321, 22mpan 689 . . . . . . . . . 10 (𝑎 No → (𝑝 ∈ ( bday 𝑎) ↔ ∃𝑞𝑎 ( bday 𝑞) = 𝑝))
2423adantr 482 . . . . . . . . 9 ((𝑎 No 𝑎 ≠ ∅) → (𝑝 ∈ ( bday 𝑎) ↔ ∃𝑞𝑎 ( bday 𝑞) = 𝑝))
2524exbidv 1925 . . . . . . . 8 ((𝑎 No 𝑎 ≠ ∅) → (∃𝑝 𝑝 ∈ ( bday 𝑎) ↔ ∃𝑝𝑞𝑎 ( bday 𝑞) = 𝑝))
2620, 25mpbird 257 . . . . . . 7 ((𝑎 No 𝑎 ≠ ∅) → ∃𝑝 𝑝 ∈ ( bday 𝑎))
27 n0 4307 . . . . . . 7 (( bday 𝑎) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ ( bday 𝑎))
2826, 27sylibr 233 . . . . . 6 ((𝑎 No 𝑎 ≠ ∅) → ( bday 𝑎) ≠ ∅)
29 onint 7726 . . . . . 6 ((( bday 𝑎) ⊆ On ∧ ( bday 𝑎) ≠ ∅) → ( bday 𝑎) ∈ ( bday 𝑎))
305, 28, 29sylancr 588 . . . . 5 ((𝑎 No 𝑎 ≠ ∅) → ( bday 𝑎) ∈ ( bday 𝑎))
31 fvelima 6909 . . . . 5 ((Fun bday ( bday 𝑎) ∈ ( bday 𝑎)) → ∃𝑝𝑎 ( bday 𝑝) = ( bday 𝑎))
322, 30, 31sylancr 588 . . . 4 ((𝑎 No 𝑎 ≠ ∅) → ∃𝑝𝑎 ( bday 𝑝) = ( bday 𝑎))
33 fnfvima 7184 . . . . . . . . . 10 (( bday Fn No 𝑎 No 𝑞𝑎) → ( bday 𝑞) ∈ ( bday 𝑎))
3421, 33mp3an1 1449 . . . . . . . . 9 ((𝑎 No 𝑞𝑎) → ( bday 𝑞) ∈ ( bday 𝑎))
3534adantlr 714 . . . . . . . 8 (((𝑎 No 𝑎 ≠ ∅) ∧ 𝑞𝑎) → ( bday 𝑞) ∈ ( bday 𝑎))
36 onnmin 7734 . . . . . . . 8 ((( bday 𝑎) ⊆ On ∧ ( bday 𝑞) ∈ ( bday 𝑎)) → ¬ ( bday 𝑞) ∈ ( bday 𝑎))
375, 35, 36sylancr 588 . . . . . . 7 (((𝑎 No 𝑎 ≠ ∅) ∧ 𝑞𝑎) → ¬ ( bday 𝑞) ∈ ( bday 𝑎))
3837ralrimiva 3144 . . . . . 6 ((𝑎 No 𝑎 ≠ ∅) → ∀𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑎))
39 eleq2 2827 . . . . . . . 8 (( bday 𝑝) = ( bday 𝑎) → (( bday 𝑞) ∈ ( bday 𝑝) ↔ ( bday 𝑞) ∈ ( bday 𝑎)))
4039notbid 318 . . . . . . 7 (( bday 𝑝) = ( bday 𝑎) → (¬ ( bday 𝑞) ∈ ( bday 𝑝) ↔ ¬ ( bday 𝑞) ∈ ( bday 𝑎)))
4140ralbidv 3175 . . . . . 6 (( bday 𝑝) = ( bday 𝑎) → (∀𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑝) ↔ ∀𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑎)))
4238, 41syl5ibrcom 247 . . . . 5 ((𝑎 No 𝑎 ≠ ∅) → (( bday 𝑝) = ( bday 𝑎) → ∀𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
4342reximdv 3168 . . . 4 ((𝑎 No 𝑎 ≠ ∅) → (∃𝑝𝑎 ( bday 𝑝) = ( bday 𝑎) → ∃𝑝𝑎𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
4432, 43mpd 15 . . 3 ((𝑎 No 𝑎 ≠ ∅) → ∃𝑝𝑎𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑝))
45 simpll 766 . . . . . . . . 9 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑎 No )
46 simprr 772 . . . . . . . . 9 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑞𝑎)
4745, 46sseldd 3946 . . . . . . . 8 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑞 No )
48 simprl 770 . . . . . . . . 9 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑝𝑎)
4945, 48sseldd 3946 . . . . . . . 8 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑝 No )
50 lrrec.1 . . . . . . . . 9 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
5150lrrecval2 27255 . . . . . . . 8 ((𝑞 No 𝑝 No ) → (𝑞𝑅𝑝 ↔ ( bday 𝑞) ∈ ( bday 𝑝)))
5247, 49, 51syl2anc 585 . . . . . . 7 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → (𝑞𝑅𝑝 ↔ ( bday 𝑞) ∈ ( bday 𝑝)))
5352notbid 318 . . . . . 6 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → (¬ 𝑞𝑅𝑝 ↔ ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
5453anassrs 469 . . . . 5 ((((𝑎 No 𝑎 ≠ ∅) ∧ 𝑝𝑎) ∧ 𝑞𝑎) → (¬ 𝑞𝑅𝑝 ↔ ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
5554ralbidva 3173 . . . 4 (((𝑎 No 𝑎 ≠ ∅) ∧ 𝑝𝑎) → (∀𝑞𝑎 ¬ 𝑞𝑅𝑝 ↔ ∀𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
5655rexbidva 3174 . . 3 ((𝑎 No 𝑎 ≠ ∅) → (∃𝑝𝑎𝑞𝑎 ¬ 𝑞𝑅𝑝 ↔ ∃𝑝𝑎𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
5744, 56mpbird 257 . 2 ((𝑎 No 𝑎 ≠ ∅) → ∃𝑝𝑎𝑞𝑎 ¬ 𝑞𝑅𝑝)
581, 57mpgbir 1802 1 𝑅 Fr No
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3446  cun 3909  wss 3911  c0 4283   cint 4908   class class class wbr 5106  {copab 5168   Fr wfr 5586  ran crn 5635  cima 5637  Oncon0 6318  Fun wfun 6491   Fn wfn 6492  cfv 6497   No csur 26991   bday cbday 26993   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-rep 5243  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-3or 1089  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-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  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-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-1o 8413  df-2o 8414  df-no 26994  df-slt 26995  df-bday 26996  df-sslt 27124  df-scut 27126  df-made 27180  df-old 27181  df-left 27183  df-right 27184
This theorem is referenced by:  noinds  27260  norecfn  27261  norecov  27262  noxpordfr  27266  no2indslem  27269  no3inds  27273
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