Theorem iunxpconst 5061

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

Assertion

Ref	Expression
iunxpconst

Distinct variable groups:   ,   ,

Proof of Theorem iunxpconst

Step	Hyp	Ref	Expression
1		xpiundir 5060	. 2
2		iunid 4385	. . 3
3	2	xpeq1i 5024	. 2
4	1, 3	eqtr3i 2488	1

Syntax hints:  =wceq 1395  {csn 4029  U_ciun 4330  X.cxp 5002

This theorem is referenced by:  ralxp  5149  rexxp  5150  mpt2mpt  6394  mpt2mpts  6864  fmpt2  6867  fsumxp  13587  fprodxp  13786  dvfval  22301  indval2  28028  filnetlem3  30198  xpiun  32454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-iun 4332  df-opab 4511  df-xp 5010