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Theorem elimasn 5019
Description: Membership in an image of a singleton. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 15-Mar-2004.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
elimasn (C (A “ {B}) ↔ B, C A)

Proof of Theorem elimasn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elex 2867 . 2 (C (A “ {B}) → C V)
2 df-br 4640 . . 3 (BACB, C A)
3 brex 4689 . . . 4 (BAC → (B V C V))
43simprd 449 . . 3 (BACC V)
52, 4sylbir 204 . 2 (B, C AC V)
6 breq2 4643 . . . 4 (x = C → (BAxBAC))
76elabg 2986 . . 3 (C V → (C {x BAx} ↔ BAC))
8 imasn 5018 . . . 4 (A “ {B}) = {x BAx}
98eleq2i 2417 . . 3 (C (A “ {B}) ↔ C {x BAx})
102bicomi 193 . . 3 (B, C ABAC)
117, 9, 103bitr4g 279 . 2 (C V → (C (A “ {B}) ↔ B, C A))
121, 5, 11pm5.21nii 342 1 (C (A “ {B}) ↔ B, C A)
Colors of variables: wff setvar class
Syntax hints:  wb 176   wcel 1710  {cab 2339  Vcvv 2859  {csn 3737  cop 4561   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:  eliniseg  5020  dfco2  5080  dfco2a  5081  fvimacnv  5403  funfvima3  5461  clos1ex  5876  elec  5964  mapexi  6003  leconnnc  6218  nmembers1lem1  6268  nchoicelem3  6291
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