Theorem eldmg 5875

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 5104	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝐴𝐵𝑦))
2	1	exbidv 1942	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦))
3		df-dm 5658	. 2 ⊢ dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦}
4	2, 3	elab2g 3640	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1561  ∃wex 1800   ∈ wcel 2143   class class class wbr 5101  dom cdm 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-dm 5658

This theorem is referenced by:  eldm2g  5876  eldm  5877  breldmg  5886  releldmb  5923  funeu  6546  fneu  6631  ndmfv  6899  erref  8699  ecdmn0  8731  rlimdm  15588  rlimdmo1  15655  iscmet3lem2  25361  dvcnp2  25989  ulmcau  26465  pserulm  26492  mulog2sum  27608  unbdqndv1  36951  eldmres  38781  eldmressnALTV  38783  eldm4  38785  eldmres2  38786  eldmcnv  38849  ssdmral  38883  eldisjdmqsim  39321  funressneu  47632  afveu  47738  rlimdmafv  47762  funressndmafv2rn  47808  afv2eu  47823  rlimdmafv2  47843  uobrcl  49805  uobeq2  50013