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Theorem iotaint 5069
Description: Equivalence between two different forms of  iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
iotaint  |-  ( E! x ph  ->  ( iota x ph )  = 
|^| { x  |  ph } )

Proof of Theorem iotaint
StepHypRef Expression
1 iotauni 5068 . 2  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 uniintabim 3776 . 2  |-  ( E! x ph  ->  U. {
x  |  ph }  =  |^| { x  | 
ph } )
31, 2eqtrd 2148 1  |-  ( E! x ph  ->  ( iota x ph )  = 
|^| { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314   E!weu 1975   {cab 2101   U.cuni 3704   |^|cint 3739   iotacio 5054
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-iota 5056
This theorem is referenced by:  bdcriota  12915
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