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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.
StepHypRef Expression
1 df-ima 4727 . . 3 (R “ {A}) = {y x {A}xRy}
2 breq1 4642 . . . . 5 (x = A → (xRyARy))
32rexsng 3766 . . . 4 (A V → (x {A}xRyARy))
43abbidv 2467 . . 3 (A V → {y x {A}xRy} = {y ARy})
51, 4syl5eq 2397 . 2 (A V → (R “ {A}) = {y ARy})
6 ima0 5013 . . 3 (R) =
7 snprc 3788 . . . . 5 A V ↔ {A} = )
87biimpi 186 . . . 4 A V → {A} = )
98imaeq2d 4942 . . 3 A V → (R “ {A}) = (R))
10 brex 4689 . . . . . . 7 (ARy → (A V y V))
1110simpld 445 . . . . . 6 (ARyA V)
1211exlimiv 1634 . . . . 5 (y ARyA V)
1312con3i 127 . . . 4 A V → ¬ y ARy)
14 abn0 3568 . . . . . 6 ({y ARy} ≠ y ARy)
15 df-ne 2518 . . . . . 6 ({y ARy} ≠ ↔ ¬ {y ARy} = )
1614, 15bitr3i 242 . . . . 5 (y ARy ↔ ¬ {y ARy} = )
1716con2bii 322 . . . 4 ({y ARy} = ↔ ¬ y ARy)
1813, 17sylibr 203 . . 3 A V → {y ARy} = )
196, 9, 183eqtr4a 2411 . 2 A V → (R “ {A}) = {y ARy})
205, 19pm2.61i 156 1 (R “ {A}) = {y ARy}
Colors of variables: wff setvar class
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-1 5  ax-2 6  ax-3 7  ax-mp 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
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