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Theorem intrnfi 7387
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 963 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  F : A --> B )
2 frn 5564 . . . 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 5562 . . . . . 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 964 . . . . 5  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  =/=  (/) )
75, 6eqnetrd 2593 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  dom  F  =/=  (/) )
8 dm0rn0 5053 . . . . 5  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
98necon3bii 2607 . . . 4  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
107, 9sylib 189 . . 3  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  ran  F  =/=  (/) )
11 simpr3 965 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  Fin )
12 ffn 5558 . . . . . 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 5626 . . . . 5  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
1513, 14sylib 189 . . . 4  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  F : A -onto-> ran  F
)
16 fofi 7359 . . . 4  |-  ( ( A  e.  Fin  /\  F : A -onto-> ran  F
)  ->  ran  F  e. 
Fin )
1711, 15, 16syl2anc 643 . . 3  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  ran  F  e.  Fin )
183, 10, 173jca 1134 . 2  |-  ( ( B  e.  V  /\  ( F : A --> B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  -> 
( ran  F  C_  B  /\  ran  F  =/=  (/)  /\  ran  F  e.  Fin ) )
19 elfir 7386 . 2  |-  ( ( B  e.  V  /\  ( ran  F  C_  B  /\  ran  F  =/=  (/)  /\  ran  F  e.  Fin ) )  ->  |^| ran  F  e.  ( fi `  B
) )
2018, 19syldan 457 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575    C_ wss 3288   (/)c0 3596   |^|cint 4018   dom cdm 4845   ran crn 4846    Fn wfn 5416   -->wf 5417   -onto->wfo 5419   ` cfv 5421   Fincfn 7076   ficfi 7381
This theorem is referenced by:  iinfi  7388  firest  13623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-1o 6691  df-er 6872  df-en 7077  df-dom 7078  df-fin 7080  df-fi 7382
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