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Mirrors > Home > MPE Home > Th. List > predep | Structured version Visualization version GIF version |
Description: The predecessor under the membership relation is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
predep | ⊢ (𝑋 ∈ 𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 6305 | . 2 ⊢ Pred( E , 𝐴, 𝑋) = (𝐴 ∩ (◡ E “ {𝑋})) | |
2 | relcnv 6108 | . . . . 5 ⊢ Rel ◡ E | |
3 | relimasn 6088 | . . . . 5 ⊢ (Rel ◡ E → (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦}) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦} |
5 | brcnvg 5882 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑋◡ E 𝑦 ↔ 𝑦 E 𝑋)) | |
6 | 5 | elvd 3478 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑋◡ E 𝑦 ↔ 𝑦 E 𝑋)) |
7 | epelg 5583 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑦 E 𝑋 ↔ 𝑦 ∈ 𝑋)) | |
8 | 6, 7 | bitrd 279 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝑋◡ E 𝑦 ↔ 𝑦 ∈ 𝑋)) |
9 | 8 | eqabcdv 2864 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → {𝑦 ∣ 𝑋◡ E 𝑦} = 𝑋) |
10 | 4, 9 | eqtrid 2780 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (◡ E “ {𝑋}) = 𝑋) |
11 | 10 | ineq2d 4212 | . 2 ⊢ (𝑋 ∈ 𝐵 → (𝐴 ∩ (◡ E “ {𝑋})) = (𝐴 ∩ 𝑋)) |
12 | 1, 11 | eqtrid 2780 | 1 ⊢ (𝑋 ∈ 𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 {cab 2705 Vcvv 3471 ∩ cin 3946 {csn 4629 class class class wbr 5148 E cep 5581 ◡ccnv 5677 “ cima 5681 Rel wrel 5683 Predcpred 6304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-eprel 5582 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 |
This theorem is referenced by: trpred 6337 predonOLD 7789 omsindsOLD 7892 |
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