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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldm4 | Structured version Visualization version GIF version |
Description: Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.) |
Ref | Expression |
---|---|
eldm4 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldmg 5551 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝐴𝑅𝑦)) | |
2 | elecALTV 34577 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑦)) | |
3 | 2 | elvd 3419 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑦)) |
4 | 3 | exbidv 2020 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦 𝑦 ∈ [𝐴]𝑅 ↔ ∃𝑦 𝐴𝑅𝑦)) |
5 | 1, 4 | bitr4d 274 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐴]𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∃wex 1878 ∈ wcel 2164 Vcvv 3414 class class class wbr 4873 dom cdm 5342 [cec 8007 |
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-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
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-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-cnv 5350 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ec 8011 |
This theorem is referenced by: eldmres2 34582 |
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