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Theorem noxpordpred 28339

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) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑅(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘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) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑅(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
2	1	xpord2pred 8162	. 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 28330	. . . . . 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 4166	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (Pred(𝑅, No, 𝐴) ∪ {𝐴}) = (((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}))
7	3	lrrecpred 28330	. . . . . 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 4166	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (Pred(𝑅, No, 𝐵) ∪ {𝐵}) = (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵}))
10	6, 9	xpeq12d 5681	. . 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 4062	. 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 2795	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∖ cdif 4045 ∪ cun 4149 {csn 4583 ⟨cop 4589 class class class wbr 5102 {copab 5166 × cxp 5648 Predcpred 6303 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000 Nocsur 27997 Lcleft 28211 Rcright 28212

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-1o 8476 df-2o 8477 df-no 28000 df-lts 28001 df-bday 28002 df-slts 28144 df-cuts 28146 df-made 28213 df-old 28214 df-left 28216 df-right 28217

This theorem is used by: norec2ov 28343

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