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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.)
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 3580 . . . . 5  |-  { x  e.  A  |  ph }  C_  A
32unissi 4263 . . . 4  |-  U. {
x  e.  A  |  ph }  C_  U. A
4 ssun2 3663 . . . 4  |-  U. A  C_  ( ~P U. A  u.  U. A )
53, 4sstri 3508 . . 3  |-  U. {
x  e.  A  |  ph }  C_  ( ~P U. A  u.  U. A
)
61, 5syl6eqss 3549 . 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 3809 . . 3  |-  (/)  C_  ( ~P U. A  u.  U. A )
97, 8syl6eqss 3549 . 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 2811   {crab 2813    u. cun 3469    C_ wss 3471   (/)c0 3780   ~Pcpw 4005   U.cuni 4240   iota_crio 6237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-sn 4023  df-pr 4025  df-uni 4241  df-iota 5544  df-riota 6238
This theorem is referenced by: (None)
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