Theorem elrn2 5883

Description: Membership in a range. (Contributed by NM, 10-Jul-1994.)

Hypothesis

Ref	Expression
elrn.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elrn2	⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elrn2

Step	Hyp	Ref	Expression
1		elrn.1	. 2 ⊢ 𝐴 ∈ V
2		elrn2g 5881	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)

Syntax hints:   ↔ wb 209  ∃wex 1806   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600  ran crn 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cnv 5670  df-dm 5672  df-rn 5673

This theorem is referenced by:  dmrnssfld  5965  rniun  6146  ssrnres  6177  fvelrn  7072  tz7.48-1  8430  prsrn  34250  dfrn5  36199  funressndmafv2rn  47883