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Theorem iotass 5073
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 5056 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 unieq 3713 . . . . . . . 8  |-  ( { x  |  ph }  =  { y }  ->  U. { x  |  ph }  =  U. { y } )
3 vex 2661 . . . . . . . . 9  |-  y  e. 
_V
43unisn 3720 . . . . . . . 8  |-  U. {
y }  =  y
52, 4syl6eq 2164 . . . . . . 7  |-  ( { x  |  ph }  =  { y }  ->  U. { x  |  ph }  =  y )
6 df-pw 3480 . . . . . . . . . . 11  |-  ~P A  =  { x  |  x 
C_  A }
76sseq2i 3092 . . . . . . . . . 10  |-  ( { x  |  ph }  C_ 
~P A  <->  { x  |  ph }  C_  { x  |  x  C_  A }
)
8 ss2ab 3133 . . . . . . . . . 10  |-  ( { x  |  ph }  C_ 
{ x  |  x 
C_  A }  <->  A. x
( ph  ->  x  C_  A ) )
97, 8bitri 183 . . . . . . . . 9  |-  ( { x  |  ph }  C_ 
~P A  <->  A. x
( ph  ->  x  C_  A ) )
109biimpri 132 . . . . . . . 8  |-  ( A. x ( ph  ->  x 
C_  A )  ->  { x  |  ph }  C_ 
~P A )
11 sspwuni 3865 . . . . . . . 8  |-  ( { x  |  ph }  C_ 
~P A  <->  U. { x  |  ph }  C_  A
)
1210, 11sylib 121 . . . . . . 7  |-  ( A. x ( ph  ->  x 
C_  A )  ->  U. { x  |  ph }  C_  A )
13 sseq1 3088 . . . . . . . 8  |-  ( U. { x  |  ph }  =  y  ->  ( U. { x  |  ph }  C_  A  <->  y  C_  A
) )
1413biimpa 292 . . . . . . 7  |-  ( ( U. { x  | 
ph }  =  y  /\  U. { x  |  ph }  C_  A
)  ->  y  C_  A )
155, 12, 14syl2anr 286 . . . . . 6  |-  ( ( A. x ( ph  ->  x  C_  A )  /\  { x  |  ph }  =  { y } )  ->  y  C_  A )
1615ex 114 . . . . 5  |-  ( A. x ( ph  ->  x 
C_  A )  -> 
( { x  | 
ph }  =  {
y }  ->  y  C_  A ) )
1716ss2abdv 3138 . . . 4  |-  ( A. x ( ph  ->  x 
C_  A )  ->  { y  |  {
x  |  ph }  =  { y } }  C_ 
{ y  |  y 
C_  A } )
18 df-pw 3480 . . . 4  |-  ~P A  =  { y  |  y 
C_  A }
1917, 18sseqtrrdi 3114 . . 3  |-  ( A. x ( ph  ->  x 
C_  A )  ->  { y  |  {
x  |  ph }  =  { y } }  C_ 
~P A )
20 sspwuni 3865 . . 3  |-  ( { y  |  { x  |  ph }  =  {
y } }  C_  ~P A  <->  U. { y  |  { x  |  ph }  =  { y } }  C_  A )
2119, 20sylib 121 . 2  |-  ( A. x ( ph  ->  x 
C_  A )  ->  U. { y  |  {
x  |  ph }  =  { y } }  C_  A )
221, 21eqsstrid 3111 1  |-  ( A. x ( ph  ->  x 
C_  A )  -> 
( iota x ph )  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312    = wceq 1314   {cab 2101    C_ wss 3039   ~Pcpw 3478   {csn 3495   U.cuni 3704   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-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-iota 5056
This theorem is referenced by:  fvss  5401  riotaexg  5700
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