Theorem noxpordpred 27949

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 8087	. 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 27940	. . . . . 6 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
5	4	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
6	5	uneq1d 4119	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (Pred(𝑅, No , 𝐴) ∪ {𝐴}) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}))
7	3	lrrecpred 27940	. . . . . 6 ⊢ (𝐵 ∈ No → Pred(𝑅, No , 𝐵) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
8	7	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑅, No , 𝐵) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
9	8	uneq1d 4119	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (Pred(𝑅, No , 𝐵) ∪ {𝐵}) = ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵}))
10	6, 9	xpeq12d 5655	. . 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 4077	. 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 2771	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 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   ∖ cdif 3898   ∪ cun 3899  {csn 4580  ⟨cop 4586   class class class wbr 5098  {copab 5160   × cxp 5622  Predcpred 6258  ‘cfv 6492  1^st c1st 7931  2^nd c2nd 7932   No csur 27607   L cleft 27821   R cright 27822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-2o 8398  df-no 27610  df-lts 27611  df-bday 27612  df-slts 27754  df-cuts 27756  df-made 27823  df-old 27824  df-left 27826  df-right 27827

This theorem is referenced by:  norec2ov  27953