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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 5769 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) | |
2 | df-br 5069 | . . 3 ⊢ (𝐴𝐵𝑦 ↔ 〈𝐴, 𝑦〉 ∈ 𝐵) | |
3 | 2 | exbii 1848 | . 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 1780 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 dom cdm 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-dm 5567 |
This theorem is referenced by: eldm2 5772 opeldmd 5777 dmfco 6759 releldm2 7744 tfrlem9 8023 climcau 15029 caucvgb 15038 lmff 21911 axhcompl-zf 28777 satfdmlem 32617 dfatdmfcoafv2 43460 |
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