Theorem iunxpconst 4784

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 4783	. 2 |- ( U_ x e. A { x } X. B ) = U_ x e. A ( { x } X. B )
2		iunid 4024	. . 3 |- U_ x e. A { x } = A
3	2	xpeq1i 4743	. 2 |- ( U_ x e. A { x } X. B ) = ( A X. B )
4	1, 3	eqtr3i 2252	1 |- U_ x e. A ( { x } X. B ) = ( A X. B )

Syntax hints:   = wceq 1395   {csn 3667   U_ciun 3968   X. cxp 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-iun 3970  df-opab 4149  df-xp 4729

This theorem is referenced by:  ralxp  4871  rexxp  4872  mpompt  6108  mpompts  6358  fmpo  6361  fsumxp  11987  fprodxp  12175  dvfvalap  15395  pwle2  16535  pwf1oexmid  16536