MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intrnfi Unicode version

Theorem intrnfi 7317
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 962 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  F : A --> B )
2 frn 5501 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2syl 15 . . 3  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  ran  F  C_  B )
4 fdm 5499 . . . . . 6  |-  ( F : A --> B  ->  dom  F  =  A )
51, 4syl 15 . . . . 5  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  dom  F  =  A )
6 simpr2 963 . . . . 5  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  =/=  (/) )
75, 6eqnetrd 2547 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  dom  F  =/=  (/) )
8 dm0rn0 4998 . . . . 5  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
98necon3bii 2561 . . . 4  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
107, 9sylib 188 . . 3  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  ran  F  =/=  (/) )
11 simpr3 964 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  Fin )
12 ffn 5495 . . . . . 6  |-  ( F : A --> B  ->  F  Fn  A )
131, 12syl 15 . . . . 5  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  F  Fn  A )
14 dffn4 5563 . . . . 5  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
1513, 14sylib 188 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  F : A -onto-> ran  F
)
16 fofi 7289 . . . 4  |-  ( ( A  e.  Fin  /\  F : A -onto-> ran  F
)  ->  ran  F  e. 
Fin )
1711, 15, 16syl2anc 642 . . 3  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  ran  F  e.  Fin )
183, 10, 173jca 1133 . 2  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  -> 
( ran  F  C_  B  /\  ran  F  =/=  (/)  /\  ran  F  e.  Fin ) )
19 elfir 7316 . 2  |-  ( ( B  e.  V  /\  ( ran  F  C_  B  /\  ran  F  =/=  (/)  /\  ran  F  e.  Fin ) )  ->  |^| ran  F  e.  ( fi `  B
) )
2018, 19syldan 456 1  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| ran  F  e.  ( fi `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529    C_ wss 3238   (/)c0 3543   |^|cint 3964   dom cdm 4792   ran crn 4793    Fn wfn 5353   -->wf 5354   -onto->wfo 5356   ` cfv 5358   Fincfn 7006   ficfi 7311
This theorem is referenced by:  iinfi  7318  firest  13547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-1o 6621  df-er 6802  df-en 7007  df-dom 7008  df-fin 7010  df-fi 7312
  Copyright terms: Public domain W3C validator