Theorem funiunfvdmf 5960

Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5959 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 2392	. . . 4 |- F/_ x z
3	1, 2	nffv 5700	. . 3 |- F/_ x ( F ` z )
4		nfcv 2392	. . 3 |- F/_ z ( F ` x )
5		fveq2 5690	. . 3 |- ( z = x -> ( F ` z ) = ( F ` x ) )
6	3, 4, 5	cbviun 4044	. 2 |- U_ z e. A ( F ` z ) = U_ x e. A ( F ` x )
7		funiunfvdm 5959	. 2 |- ( F Fn A -> U_ z e. A ( F ` z ) = U. ( F " A ) )
8	6, 7	eqtr3id 2285	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -> wi 4   = wceq 1402   F/_wnfc 2379   U.cuni 3930   U_ciun 4007   "cima 4772   Fn wfn 5367   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380

This theorem is referenced by: (None)