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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 3126 | . . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} | |
2 | vex 3417 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
3 | vex 3417 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | breldm 5565 | . . . . . . 7 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅) |
5 | 4 | adantl 475 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅) |
6 | 5 | abssi 3904 | . . . . 5 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} ⊆ dom 𝑅 |
7 | 1, 6 | eqsstri 3860 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅 |
8 | dmexg 7363 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V) | |
9 | ssexg 5031 | . . . 4 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
10 | 7, 8, 9 | sylancr 581 | . . 3 ⊢ (𝑅 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
11 | 10 | ralrimivw 3176 | . 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
12 | df-se 5306 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
13 | 11, 12 | sylibr 226 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 {cab 2811 ∀wral 3117 {crab 3121 Vcvv 3414 ⊆ wss 3798 class class class wbr 4875 Se wse 5303 dom cdm 5346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-se 5306 df-cnv 5354 df-dm 5356 df-rn 5357 |
This theorem is referenced by: dfac8clem 9175 |
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