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Theorem intrnfi 7888
Description: Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
intrnfi  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| ran  F  e.  ( fi `  B ) )

Proof of Theorem intrnfi
StepHypRef Expression
1 simpr1 1002 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  F : A --> B )
2 frn 5743 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2syl 16 . . 3  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  ran  F  C_  B )
4 fdm 5741 . . . . . 6  |-  ( F : A --> B  ->  dom  F  =  A )
51, 4syl 16 . . . . 5  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  dom  F  =  A )
6 simpr2 1003 . . . . 5  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  =/=  (/) )
75, 6eqnetrd 2760 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  dom  F  =/=  (/) )
8 dm0rn0 5225 . . . . 5  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
98necon3bii 2735 . . . 4  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
107, 9sylib 196 . . 3  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  ran  F  =/=  (/) )
11 simpr3 1004 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  Fin )
12 ffn 5737 . . . . . 6  |-  ( F : A --> B  ->  F  Fn  A )
131, 12syl 16 . . . . 5  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  F  Fn  A )
14 dffn4 5807 . . . . 5  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
1513, 14sylib 196 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  F : A -onto-> ran  F
)
16 fofi 7818 . . . 4  |-  ( ( A  e.  Fin  /\  F : A -onto-> ran  F
)  ->  ran  F  e. 
Fin )
1711, 15, 16syl2anc 661 . . 3  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  ran  F  e.  Fin )
183, 10, 173jca 1176 . 2  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  -> 
( ran  F  C_  B  /\  ran  F  =/=  (/)  /\  ran  F  e.  Fin ) )
19 elfir 7887 . 2  |-  ( ( B  e.  V  /\  ( ran  F  C_  B  /\  ran  F  =/=  (/)  /\  ran  F  e.  Fin ) )  ->  |^| ran  F  e.  ( fi `  B
) )
2018, 19syldan 470 1  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| ran  F  e.  ( fi `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3481   (/)c0 3790   |^|cint 4288   dom cdm 5005   ran crn 5006    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594   Fincfn 7528   ficfi 7882
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-fin 7532  df-fi 7883
This theorem is referenced by:  iinfi  7889  firest  14705
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