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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 6134 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | dfpred2.1 | . . . 4 ⊢ 𝑋 ∈ V | |
3 | iniseg 5946 | . . . 4 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋}) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋} |
5 | 4 | ineq2i 4174 | . 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) |
6 | 1, 5 | eqtri 2844 | 1 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {cab 2799 Vcvv 3486 ∩ cin 3923 {csn 4553 class class class wbr 5052 ◡ccnv 5540 “ cima 5544 Predcpred 6133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-xp 5547 df-cnv 5549 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 |
This theorem is referenced by: dfpred3 6144 tz6.26 6165 |
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