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Theorem ustimasn 19803
Description: Lemma for ustuqtop 19821 (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 5180 . 2  |-  ( V
" { P }
)  C_  ran  V
2 ustssxp 19779 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
323adant3 1008 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  V  C_  ( X  X.  X
) )
4 rnss 5068 . . . 4  |-  ( V 
C_  ( X  X.  X )  ->  ran  V 
C_  ran  ( X  X.  X ) )
5 rnxpid 5271 . . . 4  |-  ran  ( X  X.  X )  =  X
64, 5syl6sseq 3402 . . 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 3367 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 965    e. wcel 1756    C_ wss 3328   {csn 3877    X. cxp 4838   ran crn 4841   "cima 4843   ` cfv 5418  UnifOncust 19774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fv 5426  df-ust 19775
This theorem is referenced by:  ustuqtop0  19815  ustuqtop4  19819  utopreg  19827  ucncn  19860
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