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Theorem ustimasn 20558
Description: Lemma for ustuqtop 20576 (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
ustimasn  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ( V " { P }
)  C_  X )

Proof of Theorem ustimasn
StepHypRef Expression
1 imassrn 5348 . 2  |-  ( V
" { P }
)  C_  ran  V
2 ustssxp 20534 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
323adant3 1016 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  V  C_  ( X  X.  X
) )
4 rnss 5231 . . . 4  |-  ( V 
C_  ( X  X.  X )  ->  ran  V 
C_  ran  ( X  X.  X ) )
5 rnxpid 5440 . . . 4  |-  ran  ( X  X.  X )  =  X
64, 5syl6sseq 3550 . . 3  |-  ( V 
C_  ( X  X.  X )  ->  ran  V 
C_  X )
73, 6syl 16 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ran  V 
C_  X )
81, 7syl5ss 3515 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ( V " { P }
)  C_  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    e. wcel 1767    C_ wss 3476   {csn 4027    X. cxp 4997   ran crn 5000   "cima 5002   ` cfv 5588  UnifOncust 20529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ust 20530
This theorem is referenced by:  ustuqtop0  20570  ustuqtop4  20574  utopreg  20582  ucncn  20615
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