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Theorem intrnfi 7687
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 994 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  F : A --> B )
2 frn 5586 . . . 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 5584 . . . . . 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 995 . . . . 5  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  =/=  (/) )
75, 6eqnetrd 2654 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  dom  F  =/=  (/) )
8 dm0rn0 5077 . . . . 5  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
98necon3bii 2634 . . . 4  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
107, 9sylib 196 . . 3  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  ran  F  =/=  (/) )
11 simpr3 996 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  Fin )
12 ffn 5580 . . . . . 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 5647 . . . . 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 7618 . . . 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 1168 . 2  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  -> 
( ran  F  C_  B  /\  ran  F  =/=  (/)  /\  ran  F  e.  Fin ) )
19 elfir 7686 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620    C_ wss 3349   (/)c0 3658   |^|cint 4149   dom cdm 4861   ran crn 4862    Fn wfn 5434   -->wf 5435   -onto->wfo 5437   ` cfv 5439   Fincfn 7331   ficfi 7681
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 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-om 6498  df-1o 6941  df-er 7122  df-en 7332  df-dom 7333  df-fin 7335  df-fi 7682
This theorem is referenced by:  iinfi  7688  firest  14392
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