Theorem iunxpconst 4786

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 4785	. 2 |- ( U_ x e. A { x } X. B ) = U_ x e. A ( { x } X. B )
2		iunid 4026	. . 3 |- U_ x e. A { x } = A
3	2	xpeq1i 4745	. 2 |- ( U_ x e. A { x } X. B ) = ( A X. B )
4	1, 3	eqtr3i 2254	1 |- U_ x e. A ( { x } X. B ) = ( A X. B )

Syntax hints:   = wceq 1397   {csn 3669   U_ciun 3970   X. cxp 4723

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-iun 3972  df-opab 4151  df-xp 4731

This theorem is referenced by:  ralxp  4873  rexxp  4874  mpompt  6112  mpompts  6362  fmpo  6365  fsumxp  11996  fprodxp  12184  dvfvalap  15404  pwle2  16599  pwf1oexmid  16600