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.

Step	Hyp	Ref	Expression
1		funiunfvf.1	. . . 4 |- F/_ x F
2		nfcv 2235	. . . 4 |- F/_ x z
3	1, 2	nffv 5350	. . 3 |- F/_ x ( F ` z )
4		nfcv 2235	. . 3 |- F/_ z ( F ` x )
5		fveq2 5340	. . 3 |- ( z = x -> ( F ` z ) = ( F ` x ) )
6	3, 4, 5	cbviun 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 ) )
8	6, 7	syl5eqr 2141	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

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)