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Theorem lrrecfr 27991
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 5641 . 2 (𝑅 Fr No ↔ ∀𝑎((𝑎 No 𝑎 ≠ ∅) → ∃𝑝𝑎𝑞𝑎 ¬ 𝑞𝑅𝑝))
2 bdayfun 27832 . . . . 5 Fun bday
3 imassrn 6091 . . . . . . 7 ( bday 𝑎) ⊆ ran bday
4 bdayrn 27835 . . . . . . 7 ran bday = On
53, 4sseqtri 4032 . . . . . 6 ( bday 𝑎) ⊆ On
6 fvex 6920 . . . . . . . . . . . . 13 ( bday 𝑞) ∈ V
76jctr 524 . . . . . . . . . . . 12 (𝑞𝑎 → (𝑞𝑎 ∧ ( bday 𝑞) ∈ V))
87eximi 1832 . . . . . . . . . . 11 (∃𝑞 𝑞𝑎 → ∃𝑞(𝑞𝑎 ∧ ( bday 𝑞) ∈ V))
9 n0 4359 . . . . . . . . . . 11 (𝑎 ≠ ∅ ↔ ∃𝑞 𝑞𝑎)
10 df-rex 3069 . . . . . . . . . . 11 (∃𝑞𝑎 ( bday 𝑞) ∈ V ↔ ∃𝑞(𝑞𝑎 ∧ ( bday 𝑞) ∈ V))
118, 9, 103imtr4i 292 . . . . . . . . . 10 (𝑎 ≠ ∅ → ∃𝑞𝑎 ( bday 𝑞) ∈ V)
12 isset 3492 . . . . . . . . . . . . 13 (( bday 𝑞) ∈ V ↔ ∃𝑝 𝑝 = ( bday 𝑞))
13 eqcom 2742 . . . . . . . . . . . . . 14 (𝑝 = ( bday 𝑞) ↔ ( bday 𝑞) = 𝑝)
1413exbii 1845 . . . . . . . . . . . . 13 (∃𝑝 𝑝 = ( bday 𝑞) ↔ ∃𝑝( bday 𝑞) = 𝑝)
1512, 14bitri 275 . . . . . . . . . . . 12 (( bday 𝑞) ∈ V ↔ ∃𝑝( bday 𝑞) = 𝑝)
1615rexbii 3092 . . . . . . . . . . 11 (∃𝑞𝑎 ( bday 𝑞) ∈ V ↔ ∃𝑞𝑎𝑝( bday 𝑞) = 𝑝)
17 rexcom4 3286 . . . . . . . . . . 11 (∃𝑞𝑎𝑝( bday 𝑞) = 𝑝 ↔ ∃𝑝𝑞𝑎 ( bday 𝑞) = 𝑝)
1816, 17bitri 275 . . . . . . . . . 10 (∃𝑞𝑎 ( bday 𝑞) ∈ V ↔ ∃𝑝𝑞𝑎 ( bday 𝑞) = 𝑝)
1911, 18sylib 218 . . . . . . . . 9 (𝑎 ≠ ∅ → ∃𝑝𝑞𝑎 ( bday 𝑞) = 𝑝)
2019adantl 481 . . . . . . . 8 ((𝑎 No 𝑎 ≠ ∅) → ∃𝑝𝑞𝑎 ( bday 𝑞) = 𝑝)
21 bdayfn 27833 . . . . . . . . . . 11 bday Fn No
22 fvelimab 6981 . . . . . . . . . . 11 (( bday Fn No 𝑎 No ) → (𝑝 ∈ ( bday 𝑎) ↔ ∃𝑞𝑎 ( bday 𝑞) = 𝑝))
2321, 22mpan 690 . . . . . . . . . 10 (𝑎 No → (𝑝 ∈ ( bday 𝑎) ↔ ∃𝑞𝑎 ( bday 𝑞) = 𝑝))
2423adantr 480 . . . . . . . . 9 ((𝑎 No 𝑎 ≠ ∅) → (𝑝 ∈ ( bday 𝑎) ↔ ∃𝑞𝑎 ( bday 𝑞) = 𝑝))
2524exbidv 1919 . . . . . . . 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 4359 . . . . . . 7 (( bday 𝑎) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ ( bday 𝑎))
2826, 27sylibr 234 . . . . . 6 ((𝑎 No 𝑎 ≠ ∅) → ( bday 𝑎) ≠ ∅)
29 onint 7810 . . . . . 6 ((( bday 𝑎) ⊆ On ∧ ( bday 𝑎) ≠ ∅) → ( bday 𝑎) ∈ ( bday 𝑎))
305, 28, 29sylancr 587 . . . . 5 ((𝑎 No 𝑎 ≠ ∅) → ( bday 𝑎) ∈ ( bday 𝑎))
31 fvelima 6974 . . . . 5 ((Fun bday ( bday 𝑎) ∈ ( bday 𝑎)) → ∃𝑝𝑎 ( bday 𝑝) = ( bday 𝑎))
322, 30, 31sylancr 587 . . . 4 ((𝑎 No 𝑎 ≠ ∅) → ∃𝑝𝑎 ( bday 𝑝) = ( bday 𝑎))
33 fnfvima 7253 . . . . . . . . . 10 (( bday Fn No 𝑎 No 𝑞𝑎) → ( bday 𝑞) ∈ ( bday 𝑎))
3421, 33mp3an1 1447 . . . . . . . . 9 ((𝑎 No 𝑞𝑎) → ( bday 𝑞) ∈ ( bday 𝑎))
3534adantlr 715 . . . . . . . 8 (((𝑎 No 𝑎 ≠ ∅) ∧ 𝑞𝑎) → ( bday 𝑞) ∈ ( bday 𝑎))
36 onnmin 7818 . . . . . . . 8 ((( bday 𝑎) ⊆ On ∧ ( bday 𝑞) ∈ ( bday 𝑎)) → ¬ ( bday 𝑞) ∈ ( bday 𝑎))
375, 35, 36sylancr 587 . . . . . . 7 (((𝑎 No 𝑎 ≠ ∅) ∧ 𝑞𝑎) → ¬ ( bday 𝑞) ∈ ( bday 𝑎))
3837ralrimiva 3144 . . . . . 6 ((𝑎 No 𝑎 ≠ ∅) → ∀𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑎))
39 eleq2 2828 . . . . . . . 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 3176 . . . . . 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 767 . . . . . . . . 9 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑎 No )
46 simprr 773 . . . . . . . . 9 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑞𝑎)
4745, 46sseldd 3996 . . . . . . . 8 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑞 No )
48 simprl 771 . . . . . . . . 9 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑝𝑎)
4945, 48sseldd 3996 . . . . . . . 8 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → 𝑝 No )
50 lrrec.1 . . . . . . . . 9 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
5150lrrecval2 27988 . . . . . . . 8 ((𝑞 No 𝑝 No ) → (𝑞𝑅𝑝 ↔ ( bday 𝑞) ∈ ( bday 𝑝)))
5247, 49, 51syl2anc 584 . . . . . . 7 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → (𝑞𝑅𝑝 ↔ ( bday 𝑞) ∈ ( bday 𝑝)))
5352notbid 318 . . . . . 6 (((𝑎 No 𝑎 ≠ ∅) ∧ (𝑝𝑎𝑞𝑎)) → (¬ 𝑞𝑅𝑝 ↔ ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
5453anassrs 467 . . . . 5 ((((𝑎 No 𝑎 ≠ ∅) ∧ 𝑝𝑎) ∧ 𝑞𝑎) → (¬ 𝑞𝑅𝑝 ↔ ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
5554ralbidva 3174 . . . 4 (((𝑎 No 𝑎 ≠ ∅) ∧ 𝑝𝑎) → (∀𝑞𝑎 ¬ 𝑞𝑅𝑝 ↔ ∀𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
5655rexbidva 3175 . . 3 ((𝑎 No 𝑎 ≠ ∅) → (∃𝑝𝑎𝑞𝑎 ¬ 𝑞𝑅𝑝 ↔ ∃𝑝𝑎𝑞𝑎 ¬ ( bday 𝑞) ∈ ( bday 𝑝)))
5744, 56mpbird 257 . 2 ((𝑎 No 𝑎 ≠ ∅) → ∃𝑝𝑎𝑞𝑎 ¬ 𝑞𝑅𝑝)
581, 57mpgbir 1796 1 𝑅 Fr No
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cun 3961  wss 3963  c0 4339   cint 4951   class class class wbr 5148  {copab 5210   Fr wfr 5638  ran crn 5690  cima 5692  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  cfv 6563   No csur 27699   bday cbday 27701   L cleft 27899   R cright 27900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902  df-left 27904  df-right 27905
This theorem is referenced by:  noinds  27993  norecfn  27994  norecov  27995  noxpordfr  27999  no2indslem  28002  no3inds  28006
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