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Mirrors > Home > MPE Home > Th. List > exse2 | Structured version Visualization version GIF version |
Description: Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.) |
Ref | Expression |
---|---|
exse2 | ⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3144 | . . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} | |
2 | vex 3495 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
3 | vex 3495 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | breldm 5770 | . . . . . . 7 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅) |
5 | 4 | adantl 482 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅) |
6 | 5 | abssi 4043 | . . . . 5 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} ⊆ dom 𝑅 |
7 | 1, 6 | eqsstri 3998 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅 |
8 | dmexg 7602 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V) | |
9 | ssexg 5218 | . . . 4 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
10 | 7, 8, 9 | sylancr 587 | . . 3 ⊢ (𝑅 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
11 | 10 | ralrimivw 3180 | . 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
12 | df-se 5508 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
13 | 11, 12 | sylibr 235 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 {cab 2796 ∀wral 3135 {crab 3139 Vcvv 3492 ⊆ wss 3933 class class class wbr 5057 Se wse 5505 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-se 5508 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: dfac8clem 9446 |
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