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Theorem riotassuniOLD 6275
Description: The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.) Obsolete as of 28-Aug-2018. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
riotassuniOLD  |-  ( iota_ x  e.  A  ph )  C_  ( ~P U. A  u.  U. A )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem riotassuniOLD
StepHypRef Expression
1 riotauni 6244 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  =  U. {
x  e.  A  |  ph } )
2 ssrab2 3567 . . . . 5  |-  { x  e.  A  |  ph }  C_  A
32unissi 4253 . . . 4  |-  U. {
x  e.  A  |  ph }  C_  U. A
4 ssun2 3650 . . . 4  |-  U. A  C_  ( ~P U. A  u.  U. A )
53, 4sstri 3495 . . 3  |-  U. {
x  e.  A  |  ph }  C_  ( ~P U. A  u.  U. A
)
61, 5syl6eqss 3536 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  C_  ( ~P U. A  u.  U. A
) )
7 riotaund 6274 . . 3  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A  ph )  =  (/) )
8 0ss 3796 . . 3  |-  (/)  C_  ( ~P U. A  u.  U. A )
97, 8syl6eqss 3536 . 2  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A  ph )  C_  ( ~P U. A  u.  U. A
) )
106, 9pm2.61i 164 1  |-  ( iota_ x  e.  A  ph )  C_  ( ~P U. A  u.  U. A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   E!wreu 2793   {crab 2795    u. cun 3456    C_ wss 3458   (/)c0 3767   ~Pcpw 3993   U.cuni 4230   iota_crio 6237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-sn 4011  df-pr 4013  df-uni 4231  df-iota 5537  df-riota 6238
This theorem is referenced by: (None)
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