Theorem imasng 6082

Description: The image of a singleton. (Contributed by NM, 8-May-2005.)

Assertion

Ref	Expression
imasng	⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem imasng

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3484	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		dfima2 6060	. . 3 ⊢ (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦}
3		breq1 5126	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
4	3	rexsng 4656	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
5	4	abbidv 2800	. . 3 ⊢ (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦 ∣ 𝐴𝑅𝑦})
6	2, 5	eqtrid 2781	. 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
7	1, 6	syl 17	1 ⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  {cab 2712  ∃wrex 3059  Vcvv 3463  {csn 4606   class class class wbr 5123   “ cima 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-xp 5671  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678

This theorem is referenced by:  relimasn  6083  elimasng1  6085  args  6090  fnsnfv  6968  suppvalbr  8171  dfec2  8730  dfac3  10143  shftfib  15094  areacirclem5  37694  dfcoll2  44243  dfatsnafv2  47237