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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 4656 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝐴𝐵𝑦)) | |
2 | 1 | exbidv 1850 | . 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦)) |
3 | df-dm 5124 | . 2 ⊢ dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦} | |
4 | 2, 3 | elab2g 3353 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∃wex 1704 ∈ wcel 1990 class class class wbr 4653 dom cdm 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-dm 5124 |
This theorem is referenced by: eldm2g 5320 eldm 5321 breldmg 5330 releldmb 5360 funeu 5913 fneu 5995 ndmfv 6218 erref 7762 ecdmn0 7789 rlimdm 14282 rlimdmo1 14348 iscmet3lem2 23090 dvcnp2 23683 ulmcau 24149 pserulm 24176 mulog2sum 25226 unbdqndv1 32499 eldmres 34036 eldm4 34037 eldmres2 34038 eldmcnv 34113 afveu 41233 rlimdmafv 41257 |
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