Theorem lrrecpred 34028

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 6203	. 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 34023	. . . . 5 ⊢ ((𝑏 ∈ No ∧ 𝐴 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))))
4	3	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑏 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))))
5	4	rabbidva 3402	. . 3 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))})
6		dfrab2 4241	. . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No )
7		abid2 2881	. . . . 5 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = (( L ‘𝐴) ∪ ( R ‘𝐴))
8	7	ineq1i 4139	. . . 4 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No ) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No )
9	6, 8	eqtri 2766	. . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No )
10	5, 9	eqtrdi 2795	. 2 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ))
11		leftssno 33990	. . . . 5 ⊢ ( L ‘𝐴) ⊆ No
12	11	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ No )
13		rightssno 33991	. . . . 5 ⊢ ( R ‘𝐴) ⊆ No
14	13	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ⊆ No )
15	12, 14	unssd 4116	. . 3 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No )
16		df-ss 3900	. . 3 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No ↔ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
17	15, 16	sylib 217	. 2 ⊢ (𝐴 ∈ No → ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
18	1, 10, 17	3eqtrd 2782	1 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {cab 2715  {crab 3067   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883   class class class wbr 5070  {copab 5132  Predcpred 6190  ‘cfv 6418   No csur 33770   L cleft 33956   R cright 33957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959  df-left 33961  df-right 33962

This theorem is referenced by:  noinds  34029  norecov  34031  noxpordpred  34037  no2indslem  34038  no3inds  34042