Theorem eldmg 5879

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 5105	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝐴𝐵𝑦))
2	1	exbidv 1953	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦))
3		df-dm 5660	. 2 ⊢ dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦}
4	2, 3	elab2g 3633	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569  ∃wex 1811   ∈ wcel 2144   class class class wbr 5102  dom cdm 5650

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-dm 5660

This theorem is used by:  eldm2g  5880  eldm  5881  breldmg  5890  releldmb  5927  funeu  6565  fneu  6649  ndmfv  6917  erref  8738  ecdmn0  8770  rlimdm  15718  rlimdmo1  15785  iscmet3lem2  25613  dvcnp2  26240  ulmcau  26722  pserulm  26749  mulog2sum  27864  unbdqndv1  37416  eldmres  39252  eldmressnALTV  39254  eldm4  39256  eldmres2  39257  eldmcnv  39320  ssdmral  39354  eldisjdmqsim  39792  hfstructstruct  46068  funressneu  48160  afveu  48266  rlimdmafv  48290  funressndmafv2rn  48336  afv2eu  48351  rlimdmafv2  48371  uobrcl  50344  uobeq2  50552