Theorem iunxpconst 4830

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 4829	. 2 |- ( U_ x e. A { x } X. B ) = U_ x e. A ( { x } X. B )
2		iunid 4063	. . 3 |- U_ x e. A { x } = A
3	2	xpeq1i 4789	. 2 |- ( U_ x e. A { x } X. B ) = ( A X. B )
4	1, 3	eqtr3i 2261	1 |- U_ x e. A ( { x } X. B ) = ( A X. B )

Syntax hints:   = wceq 1402   {csn 3705   U_ciun 4007   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-iun 4009  df-opab 4188  df-xp 4775

This theorem is referenced by:  ralxp  4918  rexxp  4919  mpompt  6170  mpompts  6424  fmpo  6427  fsumxp  12181  fprodxp  12369  dvfvalap  15705  pwle2  16942  pwf1oexmid  16943