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Mirrors > Home > MPE Home > Th. List > dfpred2 | Structured version Visualization version GIF version |
Description: An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 8-Feb-2011.) |
Ref | Expression |
---|---|
dfpred2.1 | ⊢ 𝑋 ∈ V |
Ref | Expression |
---|---|
dfpred2 | ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 6254 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | dfpred2.1 | . . . 4 ⊢ 𝑋 ∈ V | |
3 | iniseg 6050 | . . . 4 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋}) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋} |
5 | 4 | ineq2i 4170 | . 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) |
6 | 1, 5 | eqtri 2761 | 1 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {cab 2710 Vcvv 3444 ∩ cin 3910 {csn 4587 class class class wbr 5106 ◡ccnv 5633 “ cima 5637 Predcpred 6253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 |
This theorem is referenced by: dfpred3 6265 tz6.26OLD 6303 |
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