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Theorem iinxprg 3722
Description: Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
iinxprg.1  C  D
iinxprg.2  C  E
Assertion
Ref Expression
iinxprg  V  W  |^|_  { ,  } C  D  i^i  E
Distinct variable groups:   ,   ,   , D   , E
Allowed substitution hints:    C()    V()    W()

Proof of Theorem iinxprg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 iinxprg.1 . . . . 5  C  D
21eleq2d 2104 . . . 4  C  D
3 iinxprg.2 . . . . 5  C  E
43eleq2d 2104 . . . 4  C  E
52, 4ralprg 3412 . . 3  V  W 
{ ,  }  C  D  E
65abbidv 2152 . 2  V  W  {  |  { ,  }  C }  {  |  D  E }
7 df-iin 3651 . 2  |^|_  { ,  } C  {  |  { ,  }  C }
8 df-in 2918 . 2  D  i^i  E  {  |  D  E }
96, 7, 83eqtr4g 2094 1  V  W  |^|_  { ,  } C  D  i^i  E
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   {cab 2023  wral 2300    i^i cin 2910   {cpr 3368   |^|_ciin 3649
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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-iin 3651
This theorem is referenced by: (None)
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