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Theorem iunxprg 3951
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 3588 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
2 iuneq1 3884 . . . 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 3950 . . 3  |-  U_ x  e.  ( { A }  u.  { B } ) C  =  ( U_ x  e.  { A } C  u.  U_ x  e.  { B } C
)
53, 4eqtri 2191 . 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 3946 . . . 4  |-  ( A  e.  V  ->  U_ x  e.  { A } C  =  D )
87adantr 274 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { A } C  =  D )
9 iunxprg.2 . . . . 5  |-  ( x  =  B  ->  C  =  E )
109iunxsng 3946 . . . 4  |-  ( B  e.  W  ->  U_ x  e.  { B } C  =  E )
1110adantl 275 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { B } C  =  E )
128, 11uneq12d 3282 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( U_ x  e. 
{ A } C  u.  U_ x  e.  { B } C )  =  ( D  u.  E
) )
135, 12eqtrid 2215 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { A ,  B } C  =  ( D  u.  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141    u. cun 3119   {csn 3581   {cpr 3582   U_ciun 3871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-sn 3587  df-pr 3588  df-iun 3873
This theorem is referenced by:  unct  12384
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