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Theorem bnj1366 28624
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 2494 . . . . . . 7  |-  F/_ y A
5 nfeu1 2219 . . . . . . 7  |-  F/ y E! y ph
64, 5nfral 2672 . . . . . 6  |-  F/ y A. x  e.  A  E! y ph
7 nfra1 2669 . . . . . . . 8  |-  F/ x A. x  e.  A  E! y ph
8 rsp 2679 . . . . . . . . . 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 5315 . . . . . . . . . 10  |-  ( E! y ph  ->  ( ph 
<->  ( iota y ph )  =  y )
)
11 eqcom 2360 . . . . . . . . . 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 2634 . . . . . . 7  |-  ( A. x  e.  A  E! y ph  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  y  =  ( iota y ph )
) )
15 abid 2346 . . . . . . 7  |-  ( y  e.  { y  |  E. x  e.  A  ph }  <->  E. x  e.  A  ph )
16 eqid 2358 . . . . . . . 8  |-  ( x  e.  A  |->  ( iota y ph ) )  =  ( x  e.  A  |->  ( iota y ph ) )
17 iotaex 5318 . . . . . . . 8  |-  ( iota y ph )  e. 
_V
1816, 17elrnmpti 5012 . . . . . . 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 1766 . . . . 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 2496 . . . . 5  |-  F/_ y { y  |  E. x  e.  A  ph }
23 nfiota1 5303 . . . . . . 7  |-  F/_ y
( iota y ph )
244, 23nfmpt 4189 . . . . . 6  |-  F/_ y
( x  e.  A  |->  ( iota y ph ) )
2524nfrn 5003 . . . . 5  |-  F/_ y ran  ( x  e.  A  |->  ( iota y ph ) )
2622, 25cleqf 2518 . . . 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 2390 . 2  |-  ( ps 
->  B  =  ran  ( x  e.  A  |->  ( iota y ph ) ) )
291simp1bi 970 . . 3  |-  ( ps 
->  A  e.  _V )
30 mptexg 5831 . . 3  |-  ( A  e.  _V  ->  (
x  e.  A  |->  ( iota y ph )
)  e.  _V )
31 rnexg 5022 . . 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 2432 1  |-  ( ps 
->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1540    = wceq 1642    e. wcel 1710   E!weu 2209   {cab 2344   A.wral 2619   E.wrex 2620   _Vcvv 2864    e. cmpt 4158   ran crn 4772   iotacio 5299
This theorem is referenced by:  bnj1489  28848
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 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345
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