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Theorem funimass1 4919
Description: A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimass1  Fun  F  C_ 
ran  F  `' F "  C_ 
C_  F "

Proof of Theorem funimass1
StepHypRef Expression
1 imass2 4644 . 2  `' F " 
C_  F " `' F "  C_  F "
2 funimacnv 4918 . . . 4  Fun 
F  F " `' F "  i^i  ran  F
3 dfss 2926 . . . . . 6 
C_  ran  F  i^i  ran  F
43biimpi 113 . . . . 5 
C_  ran  F  i^i  ran 
F
54eqcomd 2042 . . . 4 
C_  ran  F  i^i  ran  F
62, 5sylan9eq 2089 . . 3  Fun  F  C_ 
ran  F  F " `' F "
76sseq1d 2966 . 2  Fun  F  C_ 
ran  F  F " `' F "  C_  F "  C_  F "
81, 7syl5ib 143 1  Fun  F  C_ 
ran  F  `' F "  C_ 
C_  F "
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242    i^i cin 2910    C_ wss 2911   `'ccnv 4287   ran crn 4289   "cima 4291   Fun wfun 4839
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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847
This theorem is referenced by: (None)
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