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Mirrors > Home > MPE Home > Th. List > eldmg | Structured version Visualization version GIF version |
Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
eldmg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5144 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝐴𝐵𝑦)) | |
2 | 1 | exbidv 1924 | . 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦)) |
3 | df-dm 5679 | . 2 ⊢ dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦} | |
4 | 2, 3 | elab2g 3666 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∃wex 1781 ∈ wcel 2106 class class class wbr 5141 dom cdm 5669 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-dm 5679 |
This theorem is referenced by: eldm2g 5891 eldm 5892 breldmg 5901 releldmb 5937 funeu 6562 fneu 6648 ndmfv 6913 erref 8706 ecdmn0 8733 rlimdm 15477 rlimdmo1 15544 iscmet3lem2 24738 dvcnp2 25366 ulmcau 25836 pserulm 25863 mulog2sum 26967 unbdqndv1 35186 eldmres 36941 eldmressnALTV 36943 eldm4 36945 eldmres2 36946 eldmcnv 37017 funressneu 45527 afveu 45631 rlimdmafv 45655 funressndmafv2rn 45701 afv2eu 45716 rlimdmafv2 45736 |
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