Theorem eldmg 4932

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 4096	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝐴𝐵𝑦))
2	1	exbidv 1873	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦 𝐴𝐵𝑦))
3		df-dm 4741	. 2 ⊢ dom 𝐵 = {𝑥 ∣ ∃𝑦 𝑥𝐵𝑦}
4	2, 3	elab2g 2954	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2202   class class class wbr 4093  dom cdm 4731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-dm 4741

This theorem is referenced by:  eldm2g  4933  eldm  4934  breldmg  4943  releldmb  4975  funeu  5358  fneu  5443  ndmfvg  5679  erref  6765  ecdmn0m  6789  shftdm  11462  dvcnp2cntop  15510