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Theorem funiunfvf 6169
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 6168 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 2591 . . . 4  |-  F/_ x
z
31, 2nffv 5888 . . 3  |-  F/_ x
( F `  z
)
4 nfcv 2591 . . 3  |-  F/_ z
( F `  x
)
5 fveq2 5881 . . 3  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
63, 4, 5cbviun 4339 . 2  |-  U_ z  e.  A  ( F `  z )  =  U_ x  e.  A  ( F `  x )
7 funiunfv 6168 . 2  |-  ( Fun 
F  ->  U_ z  e.  A  ( F `  z )  =  U. ( F " A ) )
86, 7syl5eqr 2484 1  |-  ( Fun 
F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   F/_wnfc 2577   U.cuni 4222   U_ciun 4302   "cima 4857   Fun wfun 5595   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609
This theorem is referenced by: (None)
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