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Theorem fin 5019
Description: Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fin  F : -->  i^i  C  F : -->  F : --> C

Proof of Theorem fin
StepHypRef Expression
1 ssin 3153 . . . 4  ran  F  C_  ran  F  C_  C  ran  F  C_  i^i  C
21anbi2i 430 . . 3  F  Fn  ran  F  C_  ran  F  C_  C  F  Fn  ran  F  C_  i^i  C
3 anandi 524 . . 3  F  Fn  ran  F  C_  ran  F  C_  C  F  Fn  ran  F  C_  F  Fn  ran  F  C_  C
42, 3bitr3i 175 . 2  F  Fn  ran  F  C_  i^i  C  F  Fn  ran  F  C_  F  Fn  ran  F  C_  C
5 df-f 4849 . 2  F : -->  i^i  C  F  Fn  ran  F 
C_  i^i  C
6 df-f 4849 . . 3  F : -->  F  Fn  ran  F 
C_
7 df-f 4849 . . 3  F : --> C  F  Fn  ran  F 
C_  C
86, 7anbi12i 433 . 2  F : -->  F : --> C  F  Fn  ran  F  C_  F  Fn  ran  F  C_  C
94, 5, 83bitr4i 201 1  F : -->  i^i  C  F : -->  F : --> C
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98    i^i cin 2910    C_ wss 2911   ran crn 4289    Fn wfn 4840   -->wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-f 4849
This theorem is referenced by: (None)
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