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Theorem noxpordpred 27268
Description: Next we calculate the predecessor class 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
noxpordpred ((𝐴 No 𝐵 No ) → Pred(𝑆, ( No × No ), ⟨𝐴, 𝐵⟩) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
Distinct variable groups:   𝑥,𝑅,𝑦   𝑎,𝑏,𝐴   𝑥,𝐴,𝑦   𝐵,𝑎,𝑏   𝑥,𝐵,𝑦
Allowed substitution hints:   𝑅(𝑎,𝑏)   𝑆(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem noxpordpred
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))}
21xpord2pred 8078 . 2 ((𝐴 No 𝐵 No ) → Pred(𝑆, ( No × No ), ⟨𝐴, 𝐵⟩) = (((Pred(𝑅, No , 𝐴) ∪ {𝐴}) × (Pred(𝑅, No , 𝐵) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
3 noxpord.1 . . . . . . 7 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
43lrrecpred 27259 . . . . . 6 (𝐴 No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
54adantr 482 . . . . 5 ((𝐴 No 𝐵 No ) → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
65uneq1d 4123 . . . 4 ((𝐴 No 𝐵 No ) → (Pred(𝑅, No , 𝐴) ∪ {𝐴}) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
73lrrecpred 27259 . . . . . 6 (𝐵 No → Pred(𝑅, No , 𝐵) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
87adantl 483 . . . . 5 ((𝐴 No 𝐵 No ) → Pred(𝑅, No , 𝐵) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
98uneq1d 4123 . . . 4 ((𝐴 No 𝐵 No ) → (Pred(𝑅, No , 𝐵) ∪ {𝐵}) = ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
106, 9xpeq12d 5665 . . 3 ((𝐴 No 𝐵 No ) → ((Pred(𝑅, No , 𝐴) ∪ {𝐴}) × (Pred(𝑅, No , 𝐵) ∪ {𝐵})) = (((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})))
1110difeq1d 4082 . 2 ((𝐴 No 𝐵 No ) → (((Pred(𝑅, No , 𝐴) ∪ {𝐴}) × (Pred(𝑅, No , 𝐵) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
122, 11eqtrd 2777 1 ((𝐴 No 𝐵 No ) → Pred(𝑆, ( No × No ), ⟨𝐴, 𝐵⟩) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2944  cdif 3908  cun 3909  {csn 4587  cop 4593   class class class wbr 5106  {copab 5168   × cxp 5632  Predcpred 6253  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-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-1st 7922  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:  norec2ov  27272
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