Theorem funiunfvdmf 5915

Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5914 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.

Step	Hyp	Ref	Expression
1		funiunfvf.1	. . . 4 |- F/_ x F
2		nfcv 2375	. . . 4 |- F/_ x z
3	1, 2	nffv 5658	. . 3 |- F/_ x ( F ` z )
4		nfcv 2375	. . 3 |- F/_ z ( F ` x )
5		fveq2 5648	. . 3 |- ( z = x -> ( F ` z ) = ( F ` x ) )
6	3, 4, 5	cbviun 4012	. 2 |- U_ z e. A ( F ` z ) = U_ x e. A ( F ` x )
7		funiunfvdm 5914	. 2 |- ( F Fn A -> U_ z e. A ( F ` z ) = U. ( F " A ) )
8	6, 7	eqtr3id 2278	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -> wi 4   = wceq 1398   F/_wnfc 2362   U.cuni 3898   U_ciun 3975   "cima 4734   Fn wfn 5328   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by: (None)