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Mirrors > Home > MPE Home > Th. List > dfpred3g | Structured version Visualization version GIF version |
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 6120 | . . 3 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋)) | |
2 | breq2 5034 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑋)) | |
3 | 2 | rabbidv 3427 | . . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) |
4 | 1, 3 | eqeq12d 2814 | . 2 ⊢ (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})) |
5 | vex 3444 | . . 3 ⊢ 𝑥 ∈ V | |
6 | 5 | dfpred3 6126 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} |
7 | 4, 6 | vtoclg 3515 | 1 ⊢ (𝑋 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 Predcpred 6115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 |
This theorem is referenced by: fnrelpredd 32472 wsuclem 33225 |
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