Theorem relimasn 5946

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 4646	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		imaeq2 5919	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
3	1, 2	sylbi 219	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
4		ima0 5939	. . . . . 6 ⊢ (𝑅 “ ∅) = ∅
5	3, 4	syl6eq 2872	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅)
6	5	adantl 484	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅)
7		brrelex1 5599	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑦) → 𝐴 ∈ V)
8	7	stoic1a 1769	. . . . . 6 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦)
9	8	nexdv 1933	. . . . 5 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦)
10		abn0 4335	. . . . . 6 ⊢ ({𝑦 ∣ 𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦)
11	10	necon1bbii 3065	. . . . 5 ⊢ (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦 ∣ 𝐴𝑅𝑦} = ∅)
12	9, 11	sylib 220	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦 ∣ 𝐴𝑅𝑦} = ∅)
13	6, 12	eqtr4d 2859	. . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
14	13	ex 415	. 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}))
15		imasng 5945	. 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
16	14, 15	pm2.61d2 183	1 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799  Vcvv 3494  ∅c0 4290  {csn 4560   class class class wbr 5058   “ cima 5552  Rel wrel 5554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562

This theorem is referenced by:  elrelimasn  5947  predep  6168  fnsnfv  6737  funfv2  6745  mapsnd  8444  fnimasnd  39114  nznngen  40641  nzss  40642  hashnzfz  40645