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Mirrors > Home > MPE Home > Th. List > eldm2g | Structured version Visualization version GIF version |
Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
eldm2g | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldmg 5770 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) | |
2 | df-br 5070 | . . 3 ⊢ (𝐴𝐵𝑦 ↔ 〈𝐴, 𝑦〉 ∈ 𝐵) | |
3 | 2 | exbii 1847 | . 2 ⊢ (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵) |
4 | 1, 3 | syl6bb 289 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃wex 1779 ∈ wcel 2113 〈cop 4576 class class class wbr 5069 dom cdm 5558 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-dm 5568 |
This theorem is referenced by: eldm2 5773 opeldmd 5778 dmfco 6760 releldm2 7745 tfrlem9 8024 climcau 15030 caucvgb 15039 lmff 21912 axhcompl-zf 28778 satfdmlem 32619 dfatdmfcoafv2 43460 |
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