Theorem eldmg 5896

Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
eldmg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem eldmg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5147	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝐴𝐵𝑦))
2	1	exbidv 1917	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦))
3		df-dm 5683	. 2 ⊢ dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦}
4	2, 3	elab2g 3668	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534  ∃wex 1774   ∈ wcel 2099   class class class wbr 5144  dom cdm 5673

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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-dm 5683

This theorem is referenced by:  eldm2g  5897  eldm  5898  breldmg  5907  releldmb  5943  funeu  6574  fneu  6660  ndmfv  6926  erref  8744  ecdmn0  8773  rlimdm  15546  rlimdmo1  15613  iscmet3lem2  25306  dvcnp2  25935  dvcnp2OLD  25936  ulmcau  26419  pserulm  26446  mulog2sum  27561  unbdqndv1  36222  eldmres  37979  eldmressnALTV  37981  eldm4  37983  eldmres2  37984  eldmcnv  38054  funressneu  46696  afveu  46800  rlimdmafv  46824  funressndmafv2rn  46870  afv2eu  46885  rlimdmafv2  46905