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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 5146 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝐴𝐵𝑦)) | |
| 2 | 1 | exbidv 1921 | . 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦)) |
| 3 | df-dm 5695 | . 2 ⊢ dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦} | |
| 4 | 2, 3 | elab2g 3680 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 class class class wbr 5143 dom cdm 5685 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 |
| This theorem is referenced by: eldm2g 5910 eldm 5911 breldmg 5920 releldmb 5957 funeu 6591 fneu 6678 ndmfv 6941 erref 8765 ecdmn0 8794 rlimdm 15587 rlimdmo1 15654 iscmet3lem2 25326 dvcnp2 25955 dvcnp2OLD 25956 ulmcau 26438 pserulm 26465 mulog2sum 27581 unbdqndv1 36509 eldmres 38271 eldmressnALTV 38273 eldm4 38275 eldmres2 38276 eldmcnv 38346 funressneu 47059 afveu 47165 rlimdmafv 47189 funressndmafv2rn 47235 afv2eu 47250 rlimdmafv2 47270 |
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