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Theorem funiunfvdmf 5581
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5580 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 2235 . . . 4  |-  F/_ x
z
31, 2nffv 5350 . . 3  |-  F/_ x
( F `  z
)
4 nfcv 2235 . . 3  |-  F/_ z
( F `  x
)
5 fveq2 5340 . . 3  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
63, 4, 5cbviun 3789 . 2  |-  U_ z  e.  A  ( F `  z )  =  U_ x  e.  A  ( F `  x )
7 funiunfvdm 5580 . 2  |-  ( F  Fn  A  ->  U_ z  e.  A  ( F `  z )  =  U. ( F " A ) )
86, 7syl5eqr 2141 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296   F/_wnfc 2222   U.cuni 3675   U_ciun 3752   "cima 4470    Fn wfn 5044   ` cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-fv 5057
This theorem is referenced by: (None)
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