Theorem relimasn 6082

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 4677	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		imaeq2 6047	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
3	1, 2	sylbi 220	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
4		ima0 6074	. . . . . 6 ⊢ (𝑅 “ ∅) = ∅
5	3, 4	eqtrdi 2811	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅)
6	5	adantl 487	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅)
7		brrelex1 5703	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝐴 ∈ V)
8	7	stoic1a 1804	. . . . . 6 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑥)
9	8	alrimiv 1959	. . . . 5 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ∀𝑥 ¬ 𝐴𝑅𝑥)
10		breq2 5106	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝑥))
11	10	ab0w 4327	. . . . 5 ⊢ ({𝑦 ∣ 𝐴𝑅𝑦} = ∅ ↔ ∀𝑥 ¬ 𝐴𝑅𝑥)
12	9, 11	sylibr 237	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦 ∣ 𝐴𝑅𝑦} = ∅)
13	6, 12	eqtr4d 2798	. . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
14	13	ex 418	. 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}))
15		imasng 6081	. 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
16	14, 15	pm2.61d2 183	1 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {cab 2738  Vcvv 3450  ∅c0 4278  {csn 4583   class class class wbr 5102   “ cima 5653  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663

This theorem is used by:  elrelimasn  6083  predep  6333  funfv2  6973  mapsnd  8914  nznngen  45343  nzss  45344  hashnzfz  45347