Theorem noxpordpred 28366

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) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}

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

Step	Hyp	Ref	Expression
1		noxpord.2	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (No × No) ∧ 𝑦 ∈ (No × No) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
2	1	xpord2pred 8186	. 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))}
4	3	lrrecpred 28357	. . . . . 6 ⊢ (𝐴 ∈ No → Pred(𝑅, No, 𝐴) = ((L‘𝐴) ∪ (R‘𝐴)))
5	4	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → Pred(𝑅, No, 𝐴) = ((L‘𝐴) ∪ (R‘𝐴)))
6	5	uneq1d 4169	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (Pred(𝑅, No, 𝐴) ∪ {𝐴}) = (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
7	3	lrrecpred 28357	. . . . . 6 ⊢ (𝐵 ∈ No → Pred(𝑅, No, 𝐵) = ((L‘𝐵) ∪ (R‘𝐵)))
8	7	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → Pred(𝑅, No, 𝐵) = ((L‘𝐵) ∪ (R‘𝐵)))
9	8	uneq1d 4169	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (Pred(𝑅, No, 𝐵) ∪ {𝐵}) = (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
10	6, 9	xpeq12d 5686	. . 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})))
11	10	difeq1d 4065	. 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⟩}))
12	2, 11	eqtrd 2798	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → Pred(𝑆, (No × No), ⟨𝐴, 𝐵⟩) = (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ∖ cdif 4048   ∪ cun 4152  {csn 4588  ⟨cop 4838   class class class wbr 5106  {copab 5170   × cxp 5653  Predcpred 6318  ‘cfv 6555  1^st c1st 8020  2^nd c2nd 8021  Nocsur 28024  Lcleft 28238  Rcright 28239

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-slts 28171  df-cuts 28173  df-made 28240  df-old 28241  df-left 28243  df-right 28244

This theorem is used by:  norec2ov  28370