Theorem elrn 5005

Description: Membership in a range. (Contributed by NM, 2-Apr-2004.)

Hypothesis

Ref	Expression
elrn.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elrn	⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elrn

Step	Hyp	Ref	Expression
1		elrn.1	. . 3 ⊢ 𝐴 ∈ V
2	1	elrn2 5004	. 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)
3		df-br 4115	. . 3 ⊢ (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
4	3	exbii 1654	. 2 ⊢ (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)
5	2, 4	bitr4i 187	1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)

Syntax hints:   ↔ wb 105  ∃wex 1541   ∈ wcel 2205  Vcvv 2815  ⟨cop 3697   class class class wbr 4114  ran crn 4755

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-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  dmcosseq  5034  rnco  5274  dffo4  5830  rntpos  6501  fclim  12004  dvfgg  15679