Theorem relimasn 6074

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

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snprc 4676	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		imaeq2 6045	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
3	1, 2	sylbi 219	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
4		ima0 6066	. . . . . 6 ⊢ (𝑅 “ ∅) = ∅
5	3, 4	eqtrdi 2813	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅)
6	5	adantl 485	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅)
7		brrelex1 5700	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝐴 ∈ V)
8	7	stoic1a 1792	. . . . . 6 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑥)
9	8	alrimiv 1947	. . . . 5 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ∀𝑥 ¬ 𝐴𝑅𝑥)
10		breq2 5104	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝑥))
11	10	ab0w 4332	. . . . 5 ⊢ ({𝑦 ∣ 𝐴𝑅𝑦} = ∅ ↔ ∀𝑥 ¬ 𝐴𝑅𝑥)
12	9, 11	sylibr 236	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦 ∣ 𝐴𝑅𝑦} = ∅)
13	6, 12	eqtr4d 2800	. . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
14	13	ex 416	. 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}))
15		imasng 6073	. 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
16	14, 15	pm2.61d2 182	1 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1558   = wceq 1560   ∈ wcel 2142  {cab 2740  Vcvv 3454  ∅c0 4285  {csn 4582   class class class wbr 5100   “ cima 5650  Rel wrel 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is referenced by:  elrelimasn  6075  predep  6317  funfv2  6955  mapsnd  8868  nznngen  44892  nzss  44893  hashnzfz  44896