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Theorem iunrdx 25833
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 5617 . . . . . . 7  |-  ( F : A -onto-> C  ->  F : A --> C )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  F : A --> C )
43ffvelrnda 5840 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  C )
5 foelrn 5859 . . . . . 6  |-  ( ( F : A -onto-> C  /\  y  e.  C
)  ->  E. x  e.  A  y  =  ( F `  x ) )
61, 5sylan 468 . . . . 5  |-  ( (
ph  /\  y  e.  C )  ->  E. x  e.  A  y  =  ( F `  x ) )
7 iunrdx.2 . . . . . 6  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  D  =  B )
87eleq2d 2508 . . . . 5  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  ( z  e.  D  <->  z  e.  B
) )
94, 6, 8rexxfrd 4504 . . . 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 2555 . 2  |-  ( ph  ->  { z  |  E. x  e.  A  z  e.  B }  =  {
z  |  E. y  e.  C  z  e.  D } )
12 df-iun 4170 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
13 df-iun 4170 . 2  |-  U_ y  e.  C  D  =  { z  |  E. y  e.  C  z  e.  D }
1411, 12, 133eqtr4g 2498 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 1364    e. wcel 1761   {cab 2427   E.wrex 2714   U_ciun 4168   -->wf 5411   -onto->wfo 5413   ` cfv 5415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fo 5421  df-fv 5423
This theorem is referenced by:  volmeas  26567
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