Theorem imasng 6036

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 3452	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		dfima2 6014	. . 3 ⊢ (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦}
3		breq1 5075	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
4	3	rexsng 4608	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
5	4	abbidv 2805	. . 3 ⊢ (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦 ∣ 𝐴𝑅𝑦})
6	2, 5	eqtrid 2786	. 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
7	1, 6	syl 17	1 ⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  {cab 2717  ∃wrex 3063  Vcvv 3431  {csn 4555   class class class wbr 5072   “ cima 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631

This theorem is referenced by:  relimasn  6037  elimasng1  6039  args  6044  fnsnfv  6906  suppvalbr  8104  dfec2  8636  dfac3  10034  shftfib  15025  areacirclem5  38079  dfcoll2  44696  dfatsnafv2  47715