Theorem imasn 5018

Description: The image of a singleton. (Contributed by set.mm contributors, 9-Jan-2015.)

Assertion

Ref	Expression
imasn	⊢ (R “ {A}) = {y ∣ ARy}

Distinct variable groups:   y,A   y,R

Proof of Theorem imasn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 4727	. . 3 ⊢ (R “ {A}) = {y ∣ ∃x ∈ {A}xRy}
2		breq1 4642	. . . . 5 ⊢ (x = A → (xRy ↔ ARy))
3	2	rexsng 3766	. . . 4 ⊢ (A ∈ V → (∃x ∈ {A}xRy ↔ ARy))
4	3	abbidv 2467	. . 3 ⊢ (A ∈ V → {y ∣ ∃x ∈ {A}xRy} = {y ∣ ARy})
5	1, 4	syl5eq 2397	. 2 ⊢ (A ∈ V → (R “ {A}) = {y ∣ ARy})
6		ima0 5013	. . 3 ⊢ (R “ ∅) = ∅
7		snprc 3788	. . . . 5 ⊢ (¬ A ∈ V ↔ {A} = ∅)
8	7	biimpi 186	. . . 4 ⊢ (¬ A ∈ V → {A} = ∅)
9	8	imaeq2d 4942	. . 3 ⊢ (¬ A ∈ V → (R “ {A}) = (R “ ∅))
10		brex 4689	. . . . . . 7 ⊢ (ARy → (A ∈ V ∧ y ∈ V))
11	10	simpld 445	. . . . . 6 ⊢ (ARy → A ∈ V)
12	11	exlimiv 1634	. . . . 5 ⊢ (∃y ARy → A ∈ V)
13	12	con3i 127	. . . 4 ⊢ (¬ A ∈ V → ¬ ∃y ARy)
14		abn0 3568	. . . . . 6 ⊢ ({y ∣ ARy} ≠ ∅ ↔ ∃y ARy)
15		df-ne 2518	. . . . . 6 ⊢ ({y ∣ ARy} ≠ ∅ ↔ ¬ {y ∣ ARy} = ∅)
16	14, 15	bitr3i 242	. . . . 5 ⊢ (∃y ARy ↔ ¬ {y ∣ ARy} = ∅)
17	16	con2bii 322	. . . 4 ⊢ ({y ∣ ARy} = ∅ ↔ ¬ ∃y ARy)
18	13, 17	sylibr 203	. . 3 ⊢ (¬ A ∈ V → {y ∣ ARy} = ∅)
19	6, 9, 18	3eqtr4a 2411	. 2 ⊢ (¬ A ∈ V → (R “ {A}) = {y ∣ ARy})
20	5, 19	pm2.61i 156	1 ⊢ (R “ {A}) = {y ∣ ARy}

Syntax hints:  ¬ wn 3  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2516  ∃wrex 2615  Vcvv 2859  ∅c0 3550  {csn 3737   class class class wbr 4639   “ cima 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788

This theorem is referenced by:  elimasn  5019  epini  5021  iniseg  5022  fnsnfv  5373  funfv2  5376  fvco2  5382  dfec2  5948  mapsn  6026  nmembers1lem1  6268  nchoicelem4  6292