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Theorem iunrdx 27092
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 5788 . . . . . . 7  |-  ( F : A -onto-> C  ->  F : A --> C )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  F : A --> C )
43ffvelrnda 6014 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  C )
5 foelrn 6033 . . . . . 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 2532 . . . . 5  |-  ( (
ph  /\  y  =  ( F `  x ) )  ->  ( z  e.  D  <->  z  e.  B
) )
94, 6, 8rexxfrd 4657 . . . 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 2598 . 2  |-  ( ph  ->  { z  |  E. x  e.  A  z  e.  B }  =  {
z  |  E. y  e.  C  z  e.  D } )
12 df-iun 4322 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
13 df-iun 4322 . 2  |-  U_ y  e.  C  D  =  { z  |  E. y  e.  C  z  e.  D }
1411, 12, 133eqtr4g 2528 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 1374    e. wcel 1762   {cab 2447   E.wrex 2810   U_ciun 4320   -->wf 5577   -onto->wfo 5579   ` cfv 5581
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 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589
This theorem is referenced by:  volmeas  27831
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