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Theorem iunxpconst 4819
 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 x A ({x} × B) = (A × B)
Distinct variable groups:   x,A   x,B

Proof of Theorem iunxpconst
StepHypRef Expression
1 xpiundir 4818 . 2 (x A {x} × B) = x A ({x} × B)
2 iunid 4021 . . 3 x A {x} = A
32xpeq1i 4804 . 2 (x A {x} × B) = (A × B)
41, 3eqtr3i 2375 1 x A ({x} × B) = (A × B)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  {csn 3737  ∪ciun 3969   × cxp 4770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-xp 4784 This theorem is referenced by:  ralxp  4825  rexxp  4826  mpt2mpt  5709  fmpt2  5731
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