Theorem iotaint 5307

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

Step	Hyp	Ref	Expression
1		iotauni 5306	. 2 |- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
2		uniintabim 3970	. 2 |- ( E! x ph -> U. { x | ph } = |^| { x | ph } )
3	1, 2	eqtrd 2264	1 |- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )

Syntax hints:   -> wi 4   = wceq 1398   E!weu 2079   {cab 2217   U.cuni 3898   |^|cint 3933   iotacio 5291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-iota 5293

This theorem is referenced by:  bdcriota  16582