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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 5060 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝐴𝐵𝑦)) | |
2 | 1 | exbidv 1916 | . 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦)) |
3 | df-dm 5558 | . 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 208 = wceq 1531 ∃wex 1774 ∈ wcel 2108 class class class wbr 5057 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5058 df-dm 5558 |
This theorem is referenced by: eldm2g 5761 eldm 5762 breldmg 5771 releldmb 5809 funeu 6373 fneu 6454 ndmfv 6693 erref 8301 ecdmn0 8328 rlimdm 14900 rlimdmo1 14966 iscmet3lem2 23887 dvcnp2 24509 ulmcau 24975 pserulm 25002 mulog2sum 26105 unbdqndv1 33840 eldmres 35522 eldm4 35523 eldmres2 35524 eldmcnv 35594 funressneu 43273 afveu 43343 rlimdmafv 43367 funressndmafv2rn 43413 afv2eu 43428 rlimdmafv2 43448 |
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