Theorem dfiun3g 5378

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 4552	. 2 |- ( A. x e. A B e. C -> U_ x e. A B = U. { y | E. x e. A y = B } )
2		eqid 2622	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	rnmpt 5371	. . 3 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
4	3	unieqi 4445	. 2 |- U. ran ( x e. A |-> B ) = U. { y | E. x e. A y = B }
5	1, 4	syl6eqr 2674	1 |- ( A. x e. A B e. C -> U_ x e. A B = U. ran ( x e. A |-> B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  dfiun3  5380  iunon  7436  onoviun  7440  gruiun  9621  tgiun  20783  acunirnmpt2f  29461  locfinreflem  29907  carsgclctunlem2  30381  pmeasadd  30387  saliuncl  40542  salexct3  40560  salgensscntex  40562  meadjiun  40683  omeiunle  40731  ovolval5lem2  40867