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Theorem bnj1366 28862
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 972 . . 3  |-  ( ps 
->  B  =  {
y  |  E. x  e.  A  ph } )
31simp2bi 971 . . . . 5  |-  ( ps 
->  A. x  e.  A  E! y ph )
4 nfcv 2419 . . . . . . 7  |-  F/_ y A
5 nfeu1 2153 . . . . . . 7  |-  F/ y E! y ph
64, 5nfral 2596 . . . . . 6  |-  F/ y A. x  e.  A  E! y ph
7 nfra1 2593 . . . . . . . 8  |-  F/ x A. x  e.  A  E! y ph
8 rsp 2603 . . . . . . . . . 10  |-  ( A. x  e.  A  E! y ph  ->  ( x  e.  A  ->  E! y
ph ) )
98imp 418 . . . . . . . . 9  |-  ( ( A. x  e.  A  E! y ph  /\  x  e.  A )  ->  E! y ph )
10 iota1 5233 . . . . . . . . . 10  |-  ( E! y ph  ->  ( ph 
<->  ( iota y ph )  =  y )
)
11 eqcom 2285 . . . . . . . . . 10  |-  ( ( iota y ph )  =  y  <->  y  =  ( iota y ph )
)
1210, 11syl6bb 252 . . . . . . . . 9  |-  ( E! y ph  ->  ( ph 
<->  y  =  ( iota y ph ) ) )
139, 12syl 15 . . . . . . . 8  |-  ( ( A. x  e.  A  E! y ph  /\  x  e.  A )  ->  ( ph 
<->  y  =  ( iota y ph ) ) )
147, 13rexbida 2558 . . . . . . 7  |-  ( A. x  e.  A  E! y ph  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  y  =  ( iota y ph )
) )
15 abid 2271 . . . . . . 7  |-  ( y  e.  { y  |  E. x  e.  A  ph }  <->  E. x  e.  A  ph )
16 eqid 2283 . . . . . . . 8  |-  ( x  e.  A  |->  ( iota y ph ) )  =  ( x  e.  A  |->  ( iota y ph ) )
17 iotaex 5236 . . . . . . . 8  |-  ( iota y ph )  e. 
_V
1816, 17elrnmpti 4930 . . . . . . 7  |-  ( y  e.  ran  ( x  e.  A  |->  ( iota y ph ) )  <->  E. x  e.  A  y  =  ( iota y ph ) )
1914, 15, 183bitr4g 279 . . . . . 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 1745 . . . . 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 15 . . . 4  |-  ( ps 
->  A. y ( y  e.  { y  |  E. x  e.  A  ph }  <->  y  e.  ran  ( x  e.  A  |->  ( iota y ph ) ) ) )
22 nfab1 2421 . . . . 5  |-  F/_ y { y  |  E. x  e.  A  ph }
23 nfiota1 5221 . . . . . . 7  |-  F/_ y
( iota y ph )
244, 23nfmpt 4108 . . . . . 6  |-  F/_ y
( x  e.  A  |->  ( iota y ph ) )
2524nfrn 4921 . . . . 5  |-  F/_ y ran  ( x  e.  A  |->  ( iota y ph ) )
2622, 25cleqf 2443 . . . 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 203 . . 3  |-  ( ps 
->  { y  |  E. x  e.  A  ph }  =  ran  ( x  e.  A  |->  ( iota y ph ) ) )
282, 27eqtrd 2315 . 2  |-  ( ps 
->  B  =  ran  ( x  e.  A  |->  ( iota y ph ) ) )
291simp1bi 970 . . 3  |-  ( ps 
->  A  e.  _V )
30 mptexg 5745 . . 3  |-  ( A  e.  _V  ->  (
x  e.  A  |->  ( iota y ph )
)  e.  _V )
31 rnexg 4940 . . 3  |-  ( ( x  e.  A  |->  ( iota y ph )
)  e.  _V  ->  ran  ( x  e.  A  |->  ( iota y ph ) )  e.  _V )
3229, 30, 313syl 18 . 2  |-  ( ps 
->  ran  ( x  e.  A  |->  ( iota y ph ) )  e.  _V )
3328, 32eqeltrd 2357 1  |-  ( ps 
->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    e. cmpt 4077   ran crn 4690   iotacio 5217
This theorem is referenced by:  bnj1489  29086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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