Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imaiun1 Structured version   Visualization version   Unicode version

Theorem imaiun1 36314
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
Assertion
Ref Expression
imaiun1  |-  ( U_ x  e.  A  B " C )  =  U_ x  e.  A  ( B " C )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem imaiun1
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3053 . . . 4  |-  ( E. x  e.  A  E. z ( z  e.  C  /\  <. z ,  y >.  e.  B
)  <->  E. z E. x  e.  A  ( z  e.  C  /\  <. z ,  y >.  e.  B
) )
2 vex 3034 . . . . . 6  |-  y  e. 
_V
32elima3 5181 . . . . 5  |-  ( y  e.  ( B " C )  <->  E. z
( z  e.  C  /\  <. z ,  y
>.  e.  B ) )
43rexbii 2881 . . . 4  |-  ( E. x  e.  A  y  e.  ( B " C )  <->  E. x  e.  A  E. z
( z  e.  C  /\  <. z ,  y
>.  e.  B ) )
5 eliun 4274 . . . . . . 7  |-  ( <.
z ,  y >.  e.  U_ x  e.  A  B 
<->  E. x  e.  A  <. z ,  y >.  e.  B )
65anbi2i 708 . . . . . 6  |-  ( ( z  e.  C  /\  <.
z ,  y >.  e.  U_ x  e.  A  B )  <->  ( z  e.  C  /\  E. x  e.  A  <. z ,  y >.  e.  B
) )
7 r19.42v 2931 . . . . . 6  |-  ( E. x  e.  A  ( z  e.  C  /\  <.
z ,  y >.  e.  B )  <->  ( z  e.  C  /\  E. x  e.  A  <. z ,  y >.  e.  B
) )
86, 7bitr4i 260 . . . . 5  |-  ( ( z  e.  C  /\  <.
z ,  y >.  e.  U_ x  e.  A  B )  <->  E. x  e.  A  ( z  e.  C  /\  <. z ,  y >.  e.  B
) )
98exbii 1726 . . . 4  |-  ( E. z ( z  e.  C  /\  <. z ,  y >.  e.  U_ x  e.  A  B
)  <->  E. z E. x  e.  A  ( z  e.  C  /\  <. z ,  y >.  e.  B
) )
101, 4, 93bitr4ri 286 . . 3  |-  ( E. z ( z  e.  C  /\  <. z ,  y >.  e.  U_ x  e.  A  B
)  <->  E. x  e.  A  y  e.  ( B " C ) )
112elima3 5181 . . 3  |-  ( y  e.  ( U_ x  e.  A  B " C
)  <->  E. z ( z  e.  C  /\  <. z ,  y >.  e.  U_ x  e.  A  B
) )
12 eliun 4274 . . 3  |-  ( y  e.  U_ x  e.  A  ( B " C )  <->  E. x  e.  A  y  e.  ( B " C ) )
1310, 11, 123bitr4i 285 . 2  |-  ( y  e.  ( U_ x  e.  A  B " C
)  <->  y  e.  U_ x  e.  A  ( B " C ) )
1413eqriv 2468 1  |-  ( U_ x  e.  A  B " C )  =  U_ x  e.  A  ( B " C )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   E.wrex 2757   <.cop 3965   U_ciun 4269   "cima 4842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-iun 4271  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852
This theorem is referenced by:  trclimalb2  36389
  Copyright terms: Public domain W3C validator