Theorem dfiun3g 5034

Description: Alternate definition of indexed union when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
dfiun3g	|- ( A. x e. A B e. C -> U_ x e. A B = U. ran ( x e. A |-> B ) )

Proof of Theorem dfiun3g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiun2g 4039	. 2 |- ( A. x e. A B e. C -> U_ x e. A B = U. { y | E. x e. A y = B } )
2		eqid 2238	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	rnmpt 5025	. . 3 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
4	3	unieqi 3940	. 2 |- U. ran ( x e. A |-> B ) = U. { y | E. x e. A y = B }
5	1, 4	eqtr4di 2289	1 |- ( A. x e. A B e. C -> U_ x e. A B = U. ran ( x e. A |-> B ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   U.cuni 3930   U_ciun 4007   |-> cmpt 4187   ran crn 4770

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-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-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  dfiun3  5036  iunon  6545  tgiun  15097