Theorem cbviun 3963

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)

Hypotheses

Ref	Expression
cbviun.1	|- F/_ y B
cbviun.2	|- F/_ x C
cbviun.3	|- ( x = y -> B = C )

Assertion

Ref	Expression
cbviun	|- U_ x e. A B = U_ y e. A C

Distinct variable groups:   y, A   x, A

Allowed substitution hints:   B( x, y)   C( x, y)

Proof of Theorem cbviun

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviun.1	. . . . 5 |- F/_ y B
2	1	nfcri 2341	. . . 4 |- F/ y z e. B
3		cbviun.2	. . . . 5 |- F/_ x C
4	3	nfcri 2341	. . . 4 |- F/ x z e. C
5		cbviun.3	. . . . 5 |- ( x = y -> B = C )
6	5	eleq2d 2274	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
7	2, 4, 6	cbvrex 2734	. . 3 |- ( E. x e. A z e. B <-> E. y e. A z e. C )
8	7	abbii 2320	. 2 |- { z | E. x e. A z e. B } = { z | E. y e. A z e. C }
9		df-iun 3928	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
10		df-iun 3928	. 2 |- U_ y e. A C = { z | E. y e. A z e. C }
11	8, 9, 10	3eqtr4i 2235	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   {cab 2190   F/_wnfc 2334   E.wrex 2484   U_ciun 3926

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-iun 3928

This theorem is referenced by:  cbviunv  3965  funiunfvdmf  5832  mpomptsx  6282  dmmpossx  6284  fmpox  6285  fsum2dlemstep  11687  fisumcom2  11691  fsumiun  11730  fprod2dlemstep  11875  fprodcom2fi  11879  ctiunctlemf  12751  ctiunctal  12754