Theorem funiunfvdmf 5807

Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5806 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 2336	. . . 4 |- F/_ x z
3	1, 2	nffv 5564	. . 3 |- F/_ x ( F ` z )
4		nfcv 2336	. . 3 |- F/_ z ( F ` x )
5		fveq2 5554	. . 3 |- ( z = x -> ( F ` z ) = ( F ` x ) )
6	3, 4, 5	cbviun 3949	. 2 |- U_ z e. A ( F ` z ) = U_ x e. A ( F ` x )
7		funiunfvdm 5806	. 2 |- ( F Fn A -> U_ z e. A ( F ` z ) = U. ( F " A ) )
8	6, 7	eqtr3id 2240	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -> wi 4   = wceq 1364   F/_wnfc 2323   U.cuni 3835   U_ciun 3912   "cima 4662   Fn wfn 5249   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by: (None)