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Mirrors > Home > MPE Home > Th. List > lrrecpred | Structured version Visualization version GIF version |
Description: Finally, we calculate the value of the predecessor class over 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
lrrec.1 | ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} |
Ref | Expression |
---|---|
lrrecpred | ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpred3g 6311 | . 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 27661 | . . . . 5 ⊢ ((𝑏 ∈ No ∧ 𝐴 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))) |
4 | 3 | ancoms 457 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑏 ∈ No ) → (𝑏𝑅𝐴 ↔ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))) |
5 | 4 | rabbidva 3437 | . . 3 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))}) |
6 | dfrab2 4309 | . . . 4 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No ) | |
7 | abid2 2869 | . . . . 5 ⊢ {𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = (( L ‘𝐴) ∪ ( R ‘𝐴)) | |
8 | 7 | ineq1i 4207 | . . . 4 ⊢ ({𝑏 ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} ∩ No ) = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) |
9 | 6, 8 | eqtri 2758 | . . 3 ⊢ {𝑏 ∈ No ∣ 𝑏 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No ) |
10 | 5, 9 | eqtrdi 2786 | . 2 ⊢ (𝐴 ∈ No → {𝑏 ∈ No ∣ 𝑏𝑅𝐴} = ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∩ No )) |
11 | leftssno 27612 | . . . . 5 ⊢ ( L ‘𝐴) ⊆ No | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ No ) |
13 | rightssno 27613 | . . . . 5 ⊢ ( R ‘𝐴) ⊆ No | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ⊆ No ) |
15 | 12, 14 | unssd 4185 | . . 3 ⊢ (𝐴 ∈ No → (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No ) |
16 | df-ss 3964 | . . 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 2774 | 1 ⊢ (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 {cab 2707 {crab 3430 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 {copab 5209 Predcpred 6298 ‘cfv 6542 No csur 27379 L cleft 27577 R cright 27578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-1o 8468 df-2o 8469 df-no 27382 df-slt 27383 df-bday 27384 df-sslt 27519 df-scut 27521 df-made 27579 df-old 27580 df-left 27582 df-right 27583 |
This theorem is referenced by: noinds 27667 norecov 27669 noxpordpred 27675 no2indslem 27676 no3inds 27680 |
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