Theorem noxpordpred 28250

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 8148	. 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 28241	. . . . . 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 4114	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (Pred(𝑅, No , 𝐴) ∪ {𝐴}) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
7	3	lrrecpred 28241	. . . . . 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 4114	. . . 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 4073	. 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⟩}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   ∖ cdif 3896   ∪ cun 3897  {csn 4584  ⟨cop 4590   class class class wbr 5103  {copab 5167   × cxp 5653  Predcpred 6300  ‘cfv 6535  1^st c1st 7990  2^nd c2nd 7991   No csur 27908   L cleft 28122   R cright 28123

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

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 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6301  df-ord 6362  df-on 6363  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-1o 8462  df-2o 8463  df-no 27911  df-lts 27912  df-bday 27913  df-slts 28055  df-cuts 28057  df-made 28124  df-old 28125  df-left 28127  df-right 28128

This theorem is used by:  norec2ov  28254