Theorem relimasn 6036

Description: The image of a singleton. (Contributed by NM, 20-May-1998.)

Assertion

Ref	Expression
relimasn	⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅

Proof of Theorem relimasn

Step	Hyp	Ref	Expression
1		snprc 4669	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		imaeq2 6007	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
3	1, 2	sylbi 217	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
4		ima0 6028	. . . . . 6 ⊢ (𝑅 “ ∅) = ∅
5	3, 4	eqtrdi 2780	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅)
6	5	adantl 481	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅)
7		brrelex1 5672	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑦) → 𝐴 ∈ V)
8	7	stoic1a 1772	. . . . . 6 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦)
9	8	nexdv 1936	. . . . 5 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦)
10		abn0 4336	. . . . . 6 ⊢ ({𝑦 ∣ 𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦)
11	10	necon1bbii 2974	. . . . 5 ⊢ (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦 ∣ 𝐴𝑅𝑦} = ∅)
12	9, 11	sylib 218	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦 ∣ 𝐴𝑅𝑦} = ∅)
13	6, 12	eqtr4d 2767	. . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
14	13	ex 412	. 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}))
15		imasng 6035	. 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
16	14, 15	pm2.61d2 181	1 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  Vcvv 3436  ∅c0 4284  {csn 4577   class class class wbr 5092   “ cima 5622  Rel wrel 5624

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632

This theorem is referenced by:  elrelimasn  6037  predep  6278  funfv2  6911  mapsnd  8813  nznngen  44293  nzss  44294  hashnzfz  44297