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Theorem iunxprg 32676
Description: A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
Hypotheses
Ref Expression
iunxprg.1  |-  ( x  =  A  ->  C  =  D )
iunxprg.2  |-  ( x  =  B  ->  C  =  E )
Assertion
Ref Expression
iunxprg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { A ,  B } C  =  ( D  u.  E ) )
Distinct variable groups:    x, A    x, B    x, D    x, E
Allowed substitution hints:    C( x)    V( x)    W( x)

Proof of Theorem iunxprg
StepHypRef Expression
1 df-pr 4019 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
2 iuneq1 4329 . . . 4  |-  ( { A ,  B }  =  ( { A }  u.  { B } )  ->  U_ x  e.  { A ,  B } C  =  U_ x  e.  ( { A }  u.  { B } ) C )
31, 2ax-mp 5 . . 3  |-  U_ x  e.  { A ,  B } C  =  U_ x  e.  ( { A }  u.  { B } ) C
4 iunxun 4400 . . 3  |-  U_ x  e.  ( { A }  u.  { B } ) C  =  ( U_ x  e.  { A } C  u.  U_ x  e.  { B } C
)
53, 4eqtri 2483 . 2  |-  U_ x  e.  { A ,  B } C  =  ( U_ x  e.  { A } C  u.  U_ x  e.  { B } C
)
6 iunxprg.1 . . . . 5  |-  ( x  =  A  ->  C  =  D )
76iunxsng 4397 . . . 4  |-  ( A  e.  V  ->  U_ x  e.  { A } C  =  D )
87adantr 463 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { A } C  =  D )
9 iunxprg.2 . . . . 5  |-  ( x  =  B  ->  C  =  E )
109iunxsng 4397 . . . 4  |-  ( B  e.  W  ->  U_ x  e.  { B } C  =  E )
1110adantl 464 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { B } C  =  E )
128, 11uneq12d 3645 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( U_ x  e. 
{ A } C  u.  U_ x  e.  { B } C )  =  ( D  u.  E
) )
135, 12syl5eq 2507 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { A ,  B } C  =  ( D  u.  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    u. cun 3459   {csn 4016   {cpr 4018   U_ciun 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-v 3108  df-sbc 3325  df-un 3466  df-in 3468  df-ss 3475  df-sn 4017  df-pr 4019  df-iun 4317
This theorem is referenced by:  rnfdmpr  32682
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