Theorem noxpordpred 27437

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 8131	. 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 27428	. . . . . 6 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
5	4	adantr 482	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
6	5	uneq1d 4163	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (Pred(𝑅, No , 𝐴) ∪ {𝐴}) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
7	3	lrrecpred 27428	. . . . . 6 ⊢ (𝐵 ∈ No → Pred(𝑅, No , 𝐵) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
8	7	adantl 483	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑅, No , 𝐵) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
9	8	uneq1d 4163	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (Pred(𝑅, No , 𝐵) ∪ {𝐵}) = ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
10	6, 9	xpeq12d 5708	. . 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 4122	. 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 2773	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑆, ( No × No ), ⟨𝐴, 𝐵⟩) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941   ∖ cdif 3946   ∪ cun 3947  {csn 4629  ⟨cop 4635   class class class wbr 5149  {copab 5211   × cxp 5675  Predcpred 6300  ‘cfv 6544  1^st c1st 7973  2^nd c2nd 7974   No csur 27143   L cleft 27340   R cright 27341

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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-1o 8466  df-2o 8467  df-no 27146  df-slt 27147  df-bday 27148  df-sslt 27283  df-scut 27285  df-made 27342  df-old 27343  df-left 27345  df-right 27346

This theorem is referenced by:  norec2ov  27441