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Theorem bnj1366 28911
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 974 . . 3  |-  ( ps 
->  B  =  {
y  |  E. x  e.  A  ph } )
31simp2bi 973 . . . . 5  |-  ( ps 
->  A. x  e.  A  E! y ph )
4 nfcv 2544 . . . . . . 7  |-  F/_ y A
5 nfeu1 2268 . . . . . . 7  |-  F/ y E! y ph
64, 5nfral 2723 . . . . . 6  |-  F/ y A. x  e.  A  E! y ph
7 nfra1 2720 . . . . . . . 8  |-  F/ x A. x  e.  A  E! y ph
8 rsp 2730 . . . . . . . . . 10  |-  ( A. x  e.  A  E! y ph  ->  ( x  e.  A  ->  E! y
ph ) )
98imp 419 . . . . . . . . 9  |-  ( ( A. x  e.  A  E! y ph  /\  x  e.  A )  ->  E! y ph )
10 iota1 5395 . . . . . . . . . 10  |-  ( E! y ph  ->  ( ph 
<->  ( iota y ph )  =  y )
)
11 eqcom 2410 . . . . . . . . . 10  |-  ( ( iota y ph )  =  y  <->  y  =  ( iota y ph )
)
1210, 11syl6bb 253 . . . . . . . . 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 2685 . . . . . . 7  |-  ( A. x  e.  A  E! y ph  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  y  =  ( iota y ph )
) )
15 abid 2396 . . . . . . 7  |-  ( y  e.  { y  |  E. x  e.  A  ph }  <->  E. x  e.  A  ph )
16 eqid 2408 . . . . . . . 8  |-  ( x  e.  A  |->  ( iota y ph ) )  =  ( x  e.  A  |->  ( iota y ph ) )
17 iotaex 5398 . . . . . . . 8  |-  ( iota y ph )  e. 
_V
1816, 17elrnmpti 5084 . . . . . . 7  |-  ( y  e.  ran  ( x  e.  A  |->  ( iota y ph ) )  <->  E. x  e.  A  y  =  ( iota y ph ) )
1914, 15, 183bitr4g 280 . . . . . 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 1777 . . . . 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 2546 . . . . 5  |-  F/_ y { y  |  E. x  e.  A  ph }
23 nfiota1 5383 . . . . . . 7  |-  F/_ y
( iota y ph )
244, 23nfmpt 4261 . . . . . 6  |-  F/_ y
( x  e.  A  |->  ( iota y ph ) )
2524nfrn 5075 . . . . 5  |-  F/_ y ran  ( x  e.  A  |->  ( iota y ph ) )
2622, 25cleqf 2568 . . . 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 204 . . 3  |-  ( ps 
->  { y  |  E. x  e.  A  ph }  =  ran  ( x  e.  A  |->  ( iota y ph ) ) )
282, 27eqtrd 2440 . 2  |-  ( ps 
->  B  =  ran  ( x  e.  A  |->  ( iota y ph ) ) )
291simp1bi 972 . . 3  |-  ( ps 
->  A  e.  _V )
30 mptexg 5928 . . 3  |-  ( A  e.  _V  ->  (
x  e.  A  |->  ( iota y ph )
)  e.  _V )
31 rnexg 5094 . . 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 2482 1  |-  ( ps 
->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1721   E!weu 2258   {cab 2394   A.wral 2670   E.wrex 2671   _Vcvv 2920    e. cmpt 4230   ran crn 4842   iotacio 5379
This theorem is referenced by:  bnj1489  29135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425
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