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Theorem rabasiun 4319
Description: A class abstraction with a restricted existential quantification expressed as indexed union. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
Assertion
Ref Expression
rabasiun  |-  { x  e.  X  |  E. y  e.  Y  ph }  =  U_ y  e.  Y  { x  e.  X  |  ph }
Distinct variable groups:    x, X, y    x, Y, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem rabasiun
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfcv 2616 . . . . . 6  |-  F/_ z X
21nfcri 2609 . . . . 5  |-  F/ z  x  e.  X
3 nfv 1712 . . . . 5  |-  F/ z E. y  e.  Y  ph
42, 3nfan 1933 . . . 4  |-  F/ z ( x  e.  X  /\  E. y  e.  Y  ph )
5 nfcv 2616 . . . . . 6  |-  F/_ x X
65nfcri 2609 . . . . 5  |-  F/ x  z  e.  X
7 nfcv 2616 . . . . . 6  |-  F/_ x Y
8 nfs1v 2183 . . . . . 6  |-  F/ x [ z  /  x ] ph
97, 8nfrex 2917 . . . . 5  |-  F/ x E. y  e.  Y  [ z  /  x ] ph
106, 9nfan 1933 . . . 4  |-  F/ x
( z  e.  X  /\  E. y  e.  Y  [ z  /  x ] ph )
11 eleq1 2526 . . . . 5  |-  ( x  =  z  ->  (
x  e.  X  <->  z  e.  X ) )
12 sbequ12 1997 . . . . . 6  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
1312rexbidv 2965 . . . . 5  |-  ( x  =  z  ->  ( E. y  e.  Y  ph  <->  E. y  e.  Y  [
z  /  x ] ph ) )
1411, 13anbi12d 708 . . . 4  |-  ( x  =  z  ->  (
( x  e.  X  /\  E. y  e.  Y  ph )  <->  ( z  e.  X  /\  E. y  e.  Y  [ z  /  x ] ph )
) )
154, 10, 14cbvab 2595 . . 3  |-  { x  |  ( x  e.  X  /\  E. y  e.  Y  ph ) }  =  { z  |  ( z  e.  X  /\  E. y  e.  Y  [ z  /  x ] ph ) }
16 r19.42v 3009 . . . . 5  |-  ( E. y  e.  Y  ( z  e.  X  /\  [ z  /  x ] ph )  <->  ( z  e.  X  /\  E. y  e.  Y  [ z  /  x ] ph )
)
17 nfcv 2616 . . . . . . . 8  |-  F/_ x
z
1817, 5, 8, 12elrabf 3252 . . . . . . 7  |-  ( z  e.  { x  e.  X  |  ph }  <->  ( z  e.  X  /\  [ z  /  x ] ph ) )
1918bicomi 202 . . . . . 6  |-  ( ( z  e.  X  /\  [ z  /  x ] ph )  <->  z  e.  {
x  e.  X  |  ph } )
2019rexbii 2956 . . . . 5  |-  ( E. y  e.  Y  ( z  e.  X  /\  [ z  /  x ] ph )  <->  E. y  e.  Y  z  e.  { x  e.  X  |  ph }
)
2116, 20bitr3i 251 . . . 4  |-  ( ( z  e.  X  /\  E. y  e.  Y  [
z  /  x ] ph )  <->  E. y  e.  Y  z  e.  { x  e.  X  |  ph }
)
2221abbii 2588 . . 3  |-  { z  |  ( z  e.  X  /\  E. y  e.  Y  [ z  /  x ] ph ) }  =  { z  |  E. y  e.  Y  z  e.  { x  e.  X  |  ph } }
2315, 22eqtri 2483 . 2  |-  { x  |  ( x  e.  X  /\  E. y  e.  Y  ph ) }  =  { z  |  E. y  e.  Y  z  e.  { x  e.  X  |  ph } }
24 df-rab 2813 . 2  |-  { x  e.  X  |  E. y  e.  Y  ph }  =  { x  |  ( x  e.  X  /\  E. y  e.  Y  ph ) }
25 df-iun 4317 . 2  |-  U_ y  e.  Y  { x  e.  X  |  ph }  =  { z  |  E. y  e.  Y  z  e.  { x  e.  X  |  ph } }
2623, 24, 253eqtr4i 2493 1  |-  { x  e.  X  |  E. y  e.  Y  ph }  =  U_ y  e.  Y  { x  e.  X  |  ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398   [wsb 1744    e. wcel 1823   {cab 2439   E.wrex 2805   {crab 2808   U_ciun 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-iun 4317
This theorem is referenced by:  hashrabrex  13722
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