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Theorem lrrecpred 34142
Description: Finally, we calculate the value of the predecessor class over 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
Hypothesis
Ref Expression
lrrec.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
Assertion
Ref Expression
lrrecpred (𝐴 No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem lrrecpred
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 dfpred3g 6225 . 2 (𝐴 No → Pred(𝑅, No , 𝐴) = {𝑏 No 𝑏𝑅𝐴})
2 lrrec.1 . . . . . 6 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
32lrrecval 34137 . . . . 5 ((𝑏 No 𝐴 No ) → (𝑏𝑅𝐴𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))))
43ancoms 460 . . . 4 ((𝐴 No 𝑏 No ) → (𝑏𝑅𝐴𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))))
54rabbidva 3420 . . 3 (𝐴 No → {𝑏 No 𝑏𝑅𝐴} = {𝑏 No 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))})
6 dfrab2 4250 . . . 4 {𝑏 No 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ({𝑏𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No )
7 abid2 2880 . . . . 5 {𝑏𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = (( L ‘𝐴) ∪ ( R ‘𝐴))
87ineq1i 4148 . . . 4 ({𝑏𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No ) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No )
96, 8eqtri 2764 . . 3 {𝑏 No 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No )
105, 9eqtrdi 2792 . 2 (𝐴 No → {𝑏 No 𝑏𝑅𝐴} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ))
11 leftssno 34104 . . . . 5 ( L ‘𝐴) ⊆ No
1211a1i 11 . . . 4 (𝐴 No → ( L ‘𝐴) ⊆ No )
13 rightssno 34105 . . . . 5 ( R ‘𝐴) ⊆ No
1413a1i 11 . . . 4 (𝐴 No → ( R ‘𝐴) ⊆ No )
1512, 14unssd 4126 . . 3 (𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No )
16 df-ss 3909 . . 3 ((( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No ↔ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
1715, 16sylib 217 . 2 (𝐴 No → ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
181, 10, 173eqtrd 2780 1 (𝐴 No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2104  {cab 2713  {crab 3284  cun 3890  cin 3891  wss 3892   class class class wbr 5081  {copab 5143  Predcpred 6212  cfv 6454   No csur 33884   L cleft 34070   R cright 34071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-tp 4570  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5496  df-eprel 5502  df-po 5510  df-so 5511  df-fr 5551  df-we 5553  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-pred 6213  df-ord 6280  df-on 6281  df-suc 6283  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462  df-riota 7260  df-ov 7306  df-oprab 7307  df-mpo 7308  df-2nd 7860  df-frecs 8124  df-wrecs 8155  df-recs 8229  df-1o 8324  df-2o 8325  df-no 33887  df-slt 33888  df-bday 33889  df-sslt 34017  df-scut 34019  df-made 34072  df-old 34073  df-left 34075  df-right 34076
This theorem is referenced by:  noinds  34143  norecov  34145  noxpordpred  34151  no2indslem  34152  no3inds  34156
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