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Theorem iunrdx 27568
Description: Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Hypotheses
Ref Expression
iunrdx.1  |-  ( ph  ->  F : A -onto-> C
)
iunrdx.2  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  D  =  B )
Assertion
Ref Expression
iunrdx  |-  ( ph  ->  U_ x  e.  A  B  =  U_ y  e.  C  D )
Distinct variable groups:    x, y, A    y, B    x, C, y    x, D    x, F, y    ph, x, y
Allowed substitution hints:    B( x)    D( y)

Proof of Theorem iunrdx
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 iunrdx.1 . . . . . . 7  |-  ( ph  ->  F : A -onto-> C
)
2 fof 5801 . . . . . . 7  |-  ( F : A -onto-> C  ->  F : A --> C )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  F : A --> C )
43ffvelrnda 6032 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  C )
5 foelrn 6051 . . . . . 6  |-  ( ( F : A -onto-> C  /\  y  e.  C
)  ->  E. x  e.  A  y  =  ( F `  x ) )
61, 5sylan 471 . . . . 5  |-  ( (
ph  /\  y  e.  C )  ->  E. x  e.  A  y  =  ( F `  x ) )
7 iunrdx.2 . . . . . 6  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  D  =  B )
87eleq2d 2527 . . . . 5  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  ( z  e.  D  <->  z  e.  B
) )
94, 6, 8rexxfrd 4671 . . . 4  |-  ( ph  ->  ( E. y  e.  C  z  e.  D  <->  E. x  e.  A  z  e.  B ) )
109bicomd 201 . . 3  |-  ( ph  ->  ( E. x  e.  A  z  e.  B  <->  E. y  e.  C  z  e.  D ) )
1110abbidv 2593 . 2  |-  ( ph  ->  { z  |  E. x  e.  A  z  e.  B }  =  {
z  |  E. y  e.  C  z  e.  D } )
12 df-iun 4334 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
13 df-iun 4334 . 2  |-  U_ y  e.  C  D  =  { z  |  E. y  e.  C  z  e.  D }
1411, 12, 133eqtr4g 2523 1  |-  ( ph  ->  U_ x  e.  A  B  =  U_ y  e.  C  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808   U_ciun 4332   -->wf 5590   -onto->wfo 5592   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602
This theorem is referenced by:  volmeas  28364
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