Theorem iunxpconst 4466

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 4465	. 2 |- ( U_ x e. A { x } X. B ) = U_ x e. A ( { x } X. B )
2		iunid 3768	. . 3 |- U_ x e. A { x } = A
3	2	xpeq1i 4431	. 2 |- ( U_ x e. A { x } X. B ) = ( A X. B )
4	1, 3	eqtr3i 2107	1 |- U_ x e. A ( { x } X. B ) = ( A X. B )

Syntax hints:   = wceq 1287   {csn 3431   U_ciun 3713   X. cxp 4409

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-iun 3715  df-opab 3875  df-xp 4417

This theorem is referenced by:  ralxp  4547  rexxp  4548  mpt2mpt  5697  mpt2mpts  5925  fmpt2  5928