Theorem dfrn3 4964

Description: Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Assertion

Ref	Expression
dfrn3	⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfrn3

Step	Hyp	Ref	Expression
1		dfrn2 4963	. 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
2		df-br 4126	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	2	exbii 1658	. . 3 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
4	3	abbii 2354	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
5	1, 4	eqtri 2259	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}

Syntax hints:   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ⟨cop 3708   class class class wbr 4125  ran crn 4770

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  elrn2g  4965  elrn2  5019  imadmrn  5131  imassrn  5132  csbrng  5244