Theorem lrrecpred 28312

Description: Finally, we calculate the value of the predecessor class over 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)

Hypothesis

Ref	Expression
lrrec.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}

Assertion

Ref	Expression
lrrecpred	⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑅(𝑥, 𝑦)

Proof of Theorem lrrecpred

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfpred3g 6309	. 2 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = {𝑏 ∈ No ∣ 𝑏𝑅𝐴})
2		lrrec.1	. . . . . 6 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
3	2	lrrecval 28307	. . . . 5 ⊢ ((𝑏 ∈ No ∧ 𝐴 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))))
4	3	ancoms 464	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑏 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))))
5	4	rabbidva 3419	. . 3 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))})
6		dfrab2 4266	. . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No )
7		abid2 2898	. . . . 5 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = (( L ‘𝐴) ∪ ( R ‘𝐴))
8	7	ineq1i 4162	. . . 4 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No ) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No )
9	6, 8	eqtri 2784	. . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No )
10	5, 9	eqtrdi 2812	. 2 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ))
11		leftssno 28241	. . . . 5 ⊢ ( L ‘𝐴) ⊆ No
12	11	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ No )
13		rightssno 28242	. . . . 5 ⊢ ( R ‘𝐴) ⊆ No
14	13	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ⊆ No )
15	12, 14	unssd 4138	. . 3 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No )
16		dfss2 3917	. . 3 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No ↔ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
17	15, 16	sylib 221	. 2 ⊢ (𝐴 ∈ No → ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
18	1, 10, 17	3eqtrd 2800	1 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {cab 2739  {crab 3413   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103  {copab 5167  Predcpred 6296  ‘cfv 6531   No csur 27979   L cleft 28193   R cright 28194

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8460  df-2o 8461  df-no 27982  df-lts 27983  df-bday 27984  df-slts 28126  df-cuts 28128  df-made 28195  df-old 28196  df-left 28198  df-right 28199

This theorem is used by:  noinds  28313  norecov  28315  noxpordpred  28321  no2indlesm  28322  no3inds  28326