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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 6321 | . 2 ⊢ Pred( E , 𝐴, 𝑋) = (𝐴 ∩ (◡ E “ {𝑋})) | |
| 2 | relcnv 6122 | . . . . 5 ⊢ Rel ◡ E | |
| 3 | relimasn 6103 | . . . . 5 ⊢ (Rel ◡ E → (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦}) | |
| 4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦} |
| 5 | brcnvg 5890 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑋◡ E 𝑦 ↔ 𝑦 E 𝑋)) | |
| 6 | 5 | elvd 3486 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑋◡ E 𝑦 ↔ 𝑦 E 𝑋)) |
| 7 | epelg 5585 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑦 E 𝑋 ↔ 𝑦 ∈ 𝑋)) | |
| 8 | 6, 7 | bitrd 279 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝑋◡ E 𝑦 ↔ 𝑦 ∈ 𝑋)) |
| 9 | 8 | eqabcdv 2876 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → {𝑦 ∣ 𝑋◡ E 𝑦} = 𝑋) |
| 10 | 4, 9 | eqtrid 2789 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (◡ E “ {𝑋}) = 𝑋) |
| 11 | 10 | ineq2d 4220 | . 2 ⊢ (𝑋 ∈ 𝐵 → (𝐴 ∩ (◡ E “ {𝑋})) = (𝐴 ∩ 𝑋)) |
| 12 | 1, 11 | eqtrid 2789 | 1 ⊢ (𝑋 ∈ 𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∩ cin 3950 {csn 4626 class class class wbr 5143 E cep 5583 ◡ccnv 5684 “ cima 5688 Rel wrel 5690 Predcpred 6320 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 |
| This theorem is referenced by: trpred 6352 |
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