Theorem eldmg 5887

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 5109	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝐴𝐵𝑦))
2	1	exbidv 1956	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦))
3		df-dm 5665	. 2 ⊢ dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦}
4	2, 3	elab2g 3637	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2147   class class class wbr 5106  dom cdm 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-dm 5665

This theorem is used by:  eldm2g  5888  eldm  5889  breldmg  5898  releldmb  5935  funeu  6585  fneu  6671  ndmfv  6940  erref  8766  ecdmn0  8798  rlimdm  15747  rlimdmo1  15814  iscmet3lem2  25643  dvcnp2  26270  ulmcau  26752  pserulm  26779  mulog2sum  27894  unbdqndv1  37552  eldmres  39401  eldmressnALTV  39403  eldm4  39405  eldmres2  39406  eldmcnv  39469  ssdmral  39503  eldisjdmqsim  39941  hfstructstruct  46223  funressneu  48322  afveu  48428  rlimdmafv  48452  funressndmafv2rn  48498  afv2eu  48513  rlimdmafv2  48533  uobrcl  50514  uobeq2  50741