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Theorem unielxp 5742
Description: The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
unielxp  X.  C  U.  U.  X.  C

Proof of Theorem unielxp
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elxp7 5739 . 2  X.  C  _V  X.  _V  1st `  2nd `  C
2 elvvuni 4347 . . . 4  _V  X.  _V  U.
32adantr 261 . . 3  _V 
X.  _V  1st `  2nd `  C  U.
4 simprl 483 . . . . . 6 
U.  _V  X.  _V  1st `  2nd `  C  _V  X.  _V
5 eleq2 2098 . . . . . . . 8  U.  U.
6 eleq1 2097 . . . . . . . . 9  _V  X.  _V  _V  X.  _V
7 fveq2 5121 . . . . . . . . . . 11  1st `  1st `
87eleq1d 2103 . . . . . . . . . 10  1st `  1st `
9 fveq2 5121 . . . . . . . . . . 11  2nd `  2nd `
109eleq1d 2103 . . . . . . . . . 10  2nd `  C  2nd `  C
118, 10anbi12d 442 . . . . . . . . 9  1st `  2nd `  C  1st `  2nd `  C
126, 11anbi12d 442 . . . . . . . 8  _V  X.  _V  1st `  2nd `  C  _V  X.  _V  1st `  2nd `  C
135, 12anbi12d 442 . . . . . . 7  U.  _V  X.  _V  1st `  2nd `  C  U.  _V  X.  _V  1st `  2nd `  C
1413spcegv 2635 . . . . . 6  _V  X.  _V  U.  _V  X.  _V  1st `  2nd `  C  U.  _V  X.  _V  1st `  2nd `  C
154, 14mpcom 32 . . . . 5 
U.  _V  X.  _V  1st `  2nd `  C  U.  _V  X.  _V  1st `  2nd `  C
16 eluniab 3583 . . . . 5  U.  U. {  |  _V  X.  _V  1st `  2nd `  C } 
U.  _V  X.  _V  1st `  2nd `  C
1715, 16sylibr 137 . . . 4 
U.  _V  X.  _V  1st `  2nd `  C  U.  U. {  |  _V  X.  _V  1st `  2nd `  C }
18 xp2 5741 . . . . . 6  X.  C  {  _V  X.  _V  |  1st `  2nd `  C }
19 df-rab 2309 . . . . . 6  {  _V  X.  _V  |  1st `  2nd `  C }  {  |  _V  X.  _V  1st `  2nd `  C }
2018, 19eqtri 2057 . . . . 5  X.  C  {  |  _V 
X.  _V  1st `  2nd `  C }
2120unieqi 3581 . . . 4  U.  X.  C 
U. {  |  _V  X.  _V  1st `  2nd `  C }
2217, 21syl6eleqr 2128 . . 3 
U.  _V  X.  _V  1st `  2nd `  C  U.  U.  X.  C
233, 22mpancom 399 . 2  _V 
X.  _V  1st `  2nd `  C  U.  U.  X.  C
241, 23sylbi 114 1  X.  C  U.  U.  X.  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   {cab 2023   {crab 2304   _Vcvv 2551   U.cuni 3571    X. cxp 4286   ` cfv 4845   1stc1st 5707   2ndc2nd 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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