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Theorem funiunfvdmf 5673
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5672 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
Hypothesis
Ref Expression
funiunfvf.1  |-  F/_ x F
Assertion
Ref Expression
funiunfvdmf  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem funiunfvdmf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 funiunfvf.1 . . . 4  |-  F/_ x F
2 nfcv 2282 . . . 4  |-  F/_ x
z
31, 2nffv 5439 . . 3  |-  F/_ x
( F `  z
)
4 nfcv 2282 . . 3  |-  F/_ z
( F `  x
)
5 fveq2 5429 . . 3  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
63, 4, 5cbviun 3858 . 2  |-  U_ z  e.  A  ( F `  z )  =  U_ x  e.  A  ( F `  x )
7 funiunfvdm 5672 . 2  |-  ( F  Fn  A  ->  U_ z  e.  A  ( F `  z )  =  U. ( F " A ) )
86, 7syl5eqr 2187 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332   F/_wnfc 2269   U.cuni 3744   U_ciun 3821   "cima 4550    Fn wfn 5126   ` cfv 5131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139
This theorem is referenced by: (None)
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