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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 6310 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | dfpred2.1 | . . . 4 ⊢ 𝑋 ∈ V | |
3 | iniseg 6106 | . . . 4 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋}) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋} |
5 | 4 | ineq2i 4211 | . 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) |
6 | 1, 5 | eqtri 2756 | 1 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cab 2705 Vcvv 3473 ∩ cin 3948 {csn 4632 class class class wbr 5152 ◡ccnv 5681 “ cima 5685 Predcpred 6309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 |
This theorem is referenced by: dfpred3 6321 tz6.26OLD 6359 |
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