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| Description: An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.) | 
| Ref | Expression | 
|---|---|
| dfpred3g | ⊢ (𝑋 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | predeq3 6324 | . . 3 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋)) | |
| 2 | breq2 5146 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑋)) | |
| 3 | 2 | rabbidv 3443 | . . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) | 
| 4 | 1, 3 | eqeq12d 2752 | . 2 ⊢ (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})) | 
| 5 | vex 3483 | . . 3 ⊢ 𝑥 ∈ V | |
| 6 | 5 | dfpred3 6331 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} | 
| 7 | 4, 6 | vtoclg 3553 | 1 ⊢ (𝑋 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 {crab 3435 class class class wbr 5142 Predcpred 6319 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 | 
| This theorem is referenced by: lrrecpred 27978 fnrelpredd 35104 wsuclem 35827 | 
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