Theorem iunxpconst 4701

Description: Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Assertion

Ref	Expression
iunxpconst	|- U_ x e. A ( { x } X. B ) = ( A X. B )

Distinct variable groups:   x, A   x, B

Proof of Theorem iunxpconst

Step	Hyp	Ref	Expression
1		xpiundir 4700	. 2 |- ( U_ x e. A { x } X. B ) = U_ x e. A ( { x } X. B )
2		iunid 3957	. . 3 |- U_ x e. A { x } = A
3	2	xpeq1i 4661	. 2 |- ( U_ x e. A { x } X. B ) = ( A X. B )
4	1, 3	eqtr3i 2212	1 |- U_ x e. A ( { x } X. B ) = ( A X. B )

Syntax hints:   = wceq 1364   {csn 3607   U_ciun 3901   X. cxp 4639

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-iun 3903  df-opab 4080  df-xp 4647

This theorem is referenced by:  ralxp  4785  rexxp  4786  mpompt  5984  mpompts  6218  fmpo  6221  fsumxp  11471  fprodxp  11659  dvfvalap  14587  pwle2  15186  pwf1oexmid  15187