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Theorem bnj1366 29375
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1366.1  |-  ( ps  <->  ( A  e.  _V  /\  A. x  e.  A  E! y ph  /\  B  =  { y  |  E. x  e.  A  ph }
) )
Assertion
Ref Expression
bnj1366  |-  ( ps 
->  B  e.  _V )
Distinct variable group:    x, A, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    B( x, y)

Proof of Theorem bnj1366
StepHypRef Expression
1 bnj1366.1 . . . 4  |-  ( ps  <->  ( A  e.  _V  /\  A. x  e.  A  E! y ph  /\  B  =  { y  |  E. x  e.  A  ph }
) )
21simp3bi 975 . . 3  |-  ( ps 
->  B  =  {
y  |  E. x  e.  A  ph } )
31simp2bi 974 . . . . 5  |-  ( ps 
->  A. x  e.  A  E! y ph )
4 nfcv 2579 . . . . . . 7  |-  F/_ y A
5 nfeu1 2298 . . . . . . 7  |-  F/ y E! y ph
64, 5nfral 2766 . . . . . 6  |-  F/ y A. x  e.  A  E! y ph
7 nfra1 2763 . . . . . . . 8  |-  F/ x A. x  e.  A  E! y ph
8 rsp 2773 . . . . . . . . . 10  |-  ( A. x  e.  A  E! y ph  ->  ( x  e.  A  ->  E! y
ph ) )
98imp 420 . . . . . . . . 9  |-  ( ( A. x  e.  A  E! y ph  /\  x  e.  A )  ->  E! y ph )
10 iota1 5467 . . . . . . . . . 10  |-  ( E! y ph  ->  ( ph 
<->  ( iota y ph )  =  y )
)
11 eqcom 2445 . . . . . . . . . 10  |-  ( ( iota y ph )  =  y  <->  y  =  ( iota y ph )
)
1210, 11syl6bb 254 . . . . . . . . 9  |-  ( E! y ph  ->  ( ph 
<->  y  =  ( iota y ph ) ) )
139, 12syl 16 . . . . . . . 8  |-  ( ( A. x  e.  A  E! y ph  /\  x  e.  A )  ->  ( ph 
<->  y  =  ( iota y ph ) ) )
147, 13rexbida 2727 . . . . . . 7  |-  ( A. x  e.  A  E! y ph  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  y  =  ( iota y ph )
) )
15 abid 2431 . . . . . . 7  |-  ( y  e.  { y  |  E. x  e.  A  ph }  <->  E. x  e.  A  ph )
16 eqid 2443 . . . . . . . 8  |-  ( x  e.  A  |->  ( iota y ph ) )  =  ( x  e.  A  |->  ( iota y ph ) )
17 iotaex 5470 . . . . . . . 8  |-  ( iota y ph )  e. 
_V
1816, 17elrnmpti 5156 . . . . . . 7  |-  ( y  e.  ran  ( x  e.  A  |->  ( iota y ph ) )  <->  E. x  e.  A  y  =  ( iota y ph ) )
1914, 15, 183bitr4g 281 . . . . . 6  |-  ( A. x  e.  A  E! y ph  ->  ( y  e.  { y  |  E. x  e.  A  ph }  <->  y  e.  ran  ( x  e.  A  |->  ( iota y ph ) ) ) )
206, 19alrimi 1784 . . . . 5  |-  ( A. x  e.  A  E! y ph  ->  A. y
( y  e.  {
y  |  E. x  e.  A  ph }  <->  y  e.  ran  ( x  e.  A  |->  ( iota y ph ) ) ) )
213, 20syl 16 . . . 4  |-  ( ps 
->  A. y ( y  e.  { y  |  E. x  e.  A  ph }  <->  y  e.  ran  ( x  e.  A  |->  ( iota y ph ) ) ) )
22 nfab1 2581 . . . . 5  |-  F/_ y { y  |  E. x  e.  A  ph }
23 nfiota1 5455 . . . . . . 7  |-  F/_ y
( iota y ph )
244, 23nfmpt 4328 . . . . . 6  |-  F/_ y
( x  e.  A  |->  ( iota y ph ) )
2524nfrn 5147 . . . . 5  |-  F/_ y ran  ( x  e.  A  |->  ( iota y ph ) )
2622, 25cleqf 2603 . . . 4  |-  ( { y  |  E. x  e.  A  ph }  =  ran  ( x  e.  A  |->  ( iota y ph ) )  <->  A. y
( y  e.  {
y  |  E. x  e.  A  ph }  <->  y  e.  ran  ( x  e.  A  |->  ( iota y ph ) ) ) )
2721, 26sylibr 205 . . 3  |-  ( ps 
->  { y  |  E. x  e.  A  ph }  =  ran  ( x  e.  A  |->  ( iota y ph ) ) )
282, 27eqtrd 2475 . 2  |-  ( ps 
->  B  =  ran  ( x  e.  A  |->  ( iota y ph ) ) )
291simp1bi 973 . . 3  |-  ( ps 
->  A  e.  _V )
30 mptexg 6001 . . 3  |-  ( A  e.  _V  ->  (
x  e.  A  |->  ( iota y ph )
)  e.  _V )
31 rnexg 5166 . . 3  |-  ( ( x  e.  A  |->  ( iota y ph )
)  e.  _V  ->  ran  ( x  e.  A  |->  ( iota y ph ) )  e.  _V )
3229, 30, 313syl 19 . 2  |-  ( ps 
->  ran  ( x  e.  A  |->  ( iota y ph ) )  e.  _V )
3328, 32eqeltrd 2517 1  |-  ( ps 
->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1654    e. wcel 1728   E!weu 2288   {cab 2429   A.wral 2712   E.wrex 2713   _Vcvv 2965    e. cmpt 4297   ran crn 4914   iotacio 5451
This theorem is referenced by:  bnj1489  29599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497
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