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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 5731 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) | |
2 | df-br 5031 | . . 3 ⊢ (𝐴𝐵𝑦 ↔ 〈𝐴, 𝑦〉 ∈ 𝐵) | |
3 | 2 | exbii 1849 | . 2 ⊢ (∃𝑦 𝐴𝐵𝑦 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵) |
4 | 1, 3 | syl6bb 290 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦〈𝐴, 𝑦〉 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∃wex 1781 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 dom cdm 5519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-dm 5529 |
This theorem is referenced by: eldm2 5734 opeldmd 5739 dmfco 6734 releldm2 7724 tfrlem9 8004 climcau 15019 caucvgb 15028 lmff 21906 axhcompl-zf 28781 satfdmlem 32728 dfatdmfcoafv2 43810 |
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