Theorem funiunfvdmf 5743

Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5742 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 2312	. . . 4 |- F/_ x z
3	1, 2	nffv 5506	. . 3 |- F/_ x ( F ` z )
4		nfcv 2312	. . 3 |- F/_ z ( F ` x )
5		fveq2 5496	. . 3 |- ( z = x -> ( F ` z ) = ( F ` x ) )
6	3, 4, 5	cbviun 3910	. 2 |- U_ z e. A ( F ` z ) = U_ x e. A ( F ` x )
7		funiunfvdm 5742	. 2 |- ( F Fn A -> U_ z e. A ( F ` z ) = U. ( F " A ) )
8	6, 7	eqtr3id 2217	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -> wi 4   = wceq 1348   F/_wnfc 2299   U.cuni 3796   U_ciun 3873   "cima 4614   Fn wfn 5193   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by: (None)