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Theorem funiunfvf 5775
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 5774 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.)
Hypothesis
Ref Expression
funiunfvf.1  |-  F/_ x F
Assertion
Ref Expression
funiunfvf  |-  ( Fun 
F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem funiunfvf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 funiunfvf.1 . . . 4  |-  F/_ x F
2 nfcv 2419 . . . 4  |-  F/_ x
z
31, 2nffv 5532 . . 3  |-  F/_ x
( F `  z
)
4 nfcv 2419 . . 3  |-  F/_ z
( F `  x
)
5 fveq2 5525 . . 3  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
63, 4, 5cbviun 3939 . 2  |-  U_ z  e.  A  ( F `  z )  =  U_ x  e.  A  ( F `  x )
7 funiunfv 5774 . 2  |-  ( Fun 
F  ->  U_ z  e.  A  ( F `  z )  =  U. ( F " A ) )
86, 7syl5eqr 2329 1  |-  ( Fun 
F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   F/_wnfc 2406   U.cuni 3827   U_ciun 3905   "cima 4692   Fun wfun 5249   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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