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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 3407 | . . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} | |
2 | vex 3448 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
3 | vex 3448 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | breldm 5865 | . . . . . . 7 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅) |
5 | 4 | adantl 483 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅) |
6 | 5 | abssi 4028 | . . . . 5 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} ⊆ dom 𝑅 |
7 | 1, 6 | eqsstri 3979 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅 |
8 | dmexg 7841 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V) | |
9 | ssexg 5281 | . . . 4 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
10 | 7, 8, 9 | sylancr 588 | . . 3 ⊢ (𝑅 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
11 | 10 | ralrimivw 3144 | . 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
12 | df-se 5590 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
13 | 11, 12 | sylibr 233 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 {cab 2710 ∀wral 3061 {crab 3406 Vcvv 3444 ⊆ wss 3911 class class class wbr 5106 Se wse 5587 dom cdm 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-se 5590 df-cnv 5642 df-dm 5644 df-rn 5645 |
This theorem is referenced by: dfac8clem 9973 |
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