Theorem relimasn 3482

Description: The image of a singleton.

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 2495	. . . . . . 7 ⊢ (¬ A ∈ V ↔ {A} = ∅)
2		imaeq2 3459	. . . . . . 7 ⊢ ({A} = ∅ → (R “ {A}) = (R “ ∅))
3	1, 2	sylbi 206	. . . . . 6 ⊢ (¬ A ∈ V → (R “ {A}) = (R “ ∅))
4		ima0 3477	. . . . . 6 ⊢ (R “ ∅) = ∅
5	3, 4	syl6eq 1570	. . . . 5 ⊢ (¬ A ∈ V → (R “ {A}) = ∅)
6	5	adantl 397	. . . 4 ⊢ ((Rel R ⋀ ¬ A ∈ V) → (R “ {A}) = ∅)
7		brrelex 3264	. . . . . . . . 9 ⊢ ((Rel R ⋀ ARy) → A ∈ V)
8	7	ex 380	. . . . . . . 8 ⊢ (Rel R → (ARy → A ∈ V))
9	8	con3d 99	. . . . . . 7 ⊢ (Rel R → (¬ A ∈ V → ¬ ARy))
10	9	imp 357	. . . . . 6 ⊢ ((Rel R ⋀ ¬ A ∈ V) → ¬ ARy)
11	10	nexdv 1368	. . . . 5 ⊢ ((Rel R ⋀ ¬ A ∈ V) → ¬ ∃y ARy)
12		abn0 2342	. . . . . 6 ⊢ ({y∣ARy} ≠ ∅ ↔ ∃y ARy)
13	12	necon1bbii 1664	. . . . 5 ⊢ (¬ ∃y ARy ↔ {y∣ARy} = ∅)
14	11, 13	sylib 205	. . . 4 ⊢ ((Rel R ⋀ ¬ A ∈ V) → {y∣ARy} = ∅)
15	6, 14	eqtr4d 1557	. . 3 ⊢ ((Rel R ⋀ ¬ A ∈ V) → (R “ {A}) = {y∣ARy})
16	15	ex 380	. 2 ⊢ (Rel R → (¬ A ∈ V → (R “ {A}) = {y∣ARy}))
17		imasng 3481	. 2 ⊢ (A ∈ V → (R “ {A}) = {y∣ARy})
18	16, 17	pm2.61d2 135	1 ⊢ (Rel R → (R “ {A}) = {y∣ARy})

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 230   = wceq 997   ∈ wcel 999  ∃wex 1021  {cab 1509  Vcvv 1858  ∅c0 2331  {csn 2461   class class class wbr 2674   “ cima 3230  Rel wrel 3232

This theorem is referenced by:  fnsnfv 3824  funfv2 3828  mapsn 4406

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-br 2675  df-opab 2722  df-xp 3241  df-rel 3242  df-cnv 3243  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248

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