Theorem imasng 6058

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 3471	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		dfima2 6036	. . 3 ⊢ (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦}
3		breq1 5113	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
4	3	rexsng 4643	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
5	4	abbidv 2796	. . 3 ⊢ (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦 ∣ 𝐴𝑅𝑦})
6	2, 5	eqtrid 2777	. 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
7	1, 6	syl 17	1 ⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {cab 2708  ∃wrex 3054  Vcvv 3450  {csn 4592   class class class wbr 5110   “ cima 5644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654

This theorem is referenced by:  relimasn  6059  elimasng1  6061  args  6066  fnsnfv  6943  suppvalbr  8146  dfec2  8677  dfac3  10081  shftfib  15045  areacirclem5  37713  dfcoll2  44248  dfatsnafv2  47257