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Theorem funiunfvf 6076
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 6075 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 2616 . . . 4  |-  F/_ x
z
31, 2nffv 5807 . . 3  |-  F/_ x
( F `  z
)
4 nfcv 2616 . . 3  |-  F/_ z
( F `  x
)
5 fveq2 5800 . . 3  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
63, 4, 5cbviun 4316 . 2  |-  U_ z  e.  A  ( F `  z )  =  U_ x  e.  A  ( F `  x )
7 funiunfv 6075 . 2  |-  ( Fun 
F  ->  U_ z  e.  A  ( F `  z )  =  U. ( F " A ) )
86, 7syl5eqr 2509 1  |-  ( Fun 
F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   F/_wnfc 2602   U.cuni 4200   U_ciun 4280   "cima 4952   Fun wfun 5521   ` cfv 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-fv 5535
This theorem is referenced by: (None)
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