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Theorem iotass 4912
Description: Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
Assertion
Ref Expression
iotass  |-  ( A. x ( ph  ->  x 
C_  A )  -> 
( iota x ph )  C_  A )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem iotass
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-iota 4895 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 unieq 3617 . . . . . . . 8  |-  ( { x  |  ph }  =  { y }  ->  U. { x  |  ph }  =  U. { y } )
3 vex 2577 . . . . . . . . 9  |-  y  e. 
_V
43unisn 3624 . . . . . . . 8  |-  U. {
y }  =  y
52, 4syl6eq 2104 . . . . . . 7  |-  ( { x  |  ph }  =  { y }  ->  U. { x  |  ph }  =  y )
6 df-pw 3389 . . . . . . . . . . 11  |-  ~P A  =  { x  |  x 
C_  A }
76sseq2i 2998 . . . . . . . . . 10  |-  ( { x  |  ph }  C_ 
~P A  <->  { x  |  ph }  C_  { x  |  x  C_  A }
)
8 ss2ab 3036 . . . . . . . . . 10  |-  ( { x  |  ph }  C_ 
{ x  |  x 
C_  A }  <->  A. x
( ph  ->  x  C_  A ) )
97, 8bitri 177 . . . . . . . . 9  |-  ( { x  |  ph }  C_ 
~P A  <->  A. x
( ph  ->  x  C_  A ) )
109biimpri 128 . . . . . . . 8  |-  ( A. x ( ph  ->  x 
C_  A )  ->  { x  |  ph }  C_ 
~P A )
11 sspwuni 3767 . . . . . . . 8  |-  ( { x  |  ph }  C_ 
~P A  <->  U. { x  |  ph }  C_  A
)
1210, 11sylib 131 . . . . . . 7  |-  ( A. x ( ph  ->  x 
C_  A )  ->  U. { x  |  ph }  C_  A )
13 sseq1 2994 . . . . . . . 8  |-  ( U. { x  |  ph }  =  y  ->  ( U. { x  |  ph }  C_  A  <->  y  C_  A
) )
1413biimpa 284 . . . . . . 7  |-  ( ( U. { x  | 
ph }  =  y  /\  U. { x  |  ph }  C_  A
)  ->  y  C_  A )
155, 12, 14syl2anr 278 . . . . . 6  |-  ( ( A. x ( ph  ->  x  C_  A )  /\  { x  |  ph }  =  { y } )  ->  y  C_  A )
1615ex 112 . . . . 5  |-  ( A. x ( ph  ->  x 
C_  A )  -> 
( { x  | 
ph }  =  {
y }  ->  y  C_  A ) )
1716ss2abdv 3041 . . . 4  |-  ( A. x ( ph  ->  x 
C_  A )  ->  { y  |  {
x  |  ph }  =  { y } }  C_ 
{ y  |  y 
C_  A } )
18 df-pw 3389 . . . 4  |-  ~P A  =  { y  |  y 
C_  A }
1917, 18syl6sseqr 3020 . . 3  |-  ( A. x ( ph  ->  x 
C_  A )  ->  { y  |  {
x  |  ph }  =  { y } }  C_ 
~P A )
20 sspwuni 3767 . . 3  |-  ( { y  |  { x  |  ph }  =  {
y } }  C_  ~P A  <->  U. { y  |  { x  |  ph }  =  { y } }  C_  A )
2119, 20sylib 131 . 2  |-  ( A. x ( ph  ->  x 
C_  A )  ->  U. { y  |  {
x  |  ph }  =  { y } }  C_  A )
221, 21syl5eqss 3017 1  |-  ( A. x ( ph  ->  x 
C_  A )  -> 
( iota x ph )  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    = wceq 1259   {cab 2042    C_ wss 2945   ~Pcpw 3387   {csn 3403   U.cuni 3608   iotacio 4893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-iota 4895
This theorem is referenced by:  fvss  5217  riotaexg  5500
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