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Theorem iotass 4965
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 4948 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 unieq 3647 . . . . . . . 8  |-  ( { x  |  ph }  =  { y }  ->  U. { x  |  ph }  =  U. { y } )
3 vex 2618 . . . . . . . . 9  |-  y  e. 
_V
43unisn 3654 . . . . . . . 8  |-  U. {
y }  =  y
52, 4syl6eq 2133 . . . . . . 7  |-  ( { x  |  ph }  =  { y }  ->  U. { x  |  ph }  =  y )
6 df-pw 3417 . . . . . . . . . . 11  |-  ~P A  =  { x  |  x 
C_  A }
76sseq2i 3040 . . . . . . . . . 10  |-  ( { x  |  ph }  C_ 
~P A  <->  { x  |  ph }  C_  { x  |  x  C_  A }
)
8 ss2ab 3078 . . . . . . . . . 10  |-  ( { x  |  ph }  C_ 
{ x  |  x 
C_  A }  <->  A. x
( ph  ->  x  C_  A ) )
97, 8bitri 182 . . . . . . . . 9  |-  ( { x  |  ph }  C_ 
~P A  <->  A. x
( ph  ->  x  C_  A ) )
109biimpri 131 . . . . . . . 8  |-  ( A. x ( ph  ->  x 
C_  A )  ->  { x  |  ph }  C_ 
~P A )
11 sspwuni 3797 . . . . . . . 8  |-  ( { x  |  ph }  C_ 
~P A  <->  U. { x  |  ph }  C_  A
)
1210, 11sylib 120 . . . . . . 7  |-  ( A. x ( ph  ->  x 
C_  A )  ->  U. { x  |  ph }  C_  A )
13 sseq1 3036 . . . . . . . 8  |-  ( U. { x  |  ph }  =  y  ->  ( U. { x  |  ph }  C_  A  <->  y  C_  A
) )
1413biimpa 290 . . . . . . 7  |-  ( ( U. { x  | 
ph }  =  y  /\  U. { x  |  ph }  C_  A
)  ->  y  C_  A )
155, 12, 14syl2anr 284 . . . . . 6  |-  ( ( A. x ( ph  ->  x  C_  A )  /\  { x  |  ph }  =  { y } )  ->  y  C_  A )
1615ex 113 . . . . 5  |-  ( A. x ( ph  ->  x 
C_  A )  -> 
( { x  | 
ph }  =  {
y }  ->  y  C_  A ) )
1716ss2abdv 3083 . . . 4  |-  ( A. x ( ph  ->  x 
C_  A )  ->  { y  |  {
x  |  ph }  =  { y } }  C_ 
{ y  |  y 
C_  A } )
18 df-pw 3417 . . . 4  |-  ~P A  =  { y  |  y 
C_  A }
1917, 18syl6sseqr 3062 . . 3  |-  ( A. x ( ph  ->  x 
C_  A )  ->  { y  |  {
x  |  ph }  =  { y } }  C_ 
~P A )
20 sspwuni 3797 . . 3  |-  ( { y  |  { x  |  ph }  =  {
y } }  C_  ~P A  <->  U. { y  |  { x  |  ph }  =  { y } }  C_  A )
2119, 20sylib 120 . 2  |-  ( A. x ( ph  ->  x 
C_  A )  ->  U. { y  |  {
x  |  ph }  =  { y } }  C_  A )
221, 21syl5eqss 3059 1  |-  ( A. x ( ph  ->  x 
C_  A )  -> 
( iota x ph )  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1285    = wceq 1287   {cab 2071    C_ wss 2988   ~Pcpw 3415   {csn 3431   U.cuni 3638   iotacio 4946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-uni 3639  df-iota 4948
This theorem is referenced by:  fvss  5284  riotaexg  5575
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