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Theorem fconst4m 5324
Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
Assertion
Ref Expression
fconst4m  F : --> { }  F  Fn  `' F " { }
Distinct variable group:   ,
Allowed substitution hints:   ()    F()

Proof of Theorem fconst4m
StepHypRef Expression
1 fconst3m 5323 . 2  F : --> { }  F  Fn  C_  `' F " { }
2 cnvimass 4631 . . . . . 6  `' F " { }  C_  dom  F
3 fndm 4941 . . . . . 6  F  Fn  dom  F
42, 3syl5sseq 2987 . . . . 5  F  Fn  `' F " { }  C_
54biantrurd 289 . . . 4  F  Fn  C_  `' F " { }  `' F " { } 
C_  C_  `' F " { }
6 eqss 2954 . . . 4  `' F " { }  `' F " { }  C_  C_  `' F " { }
75, 6syl6bbr 187 . . 3  F  Fn  C_  `' F " { }  `' F " { }
87pm5.32i 427 . 2  F  Fn  C_  `' F " { }  F  Fn  `' F " { }
91, 8syl6bb 185 1  F : --> { }  F  Fn  `' F " { }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390    C_ wss 2911   {csn 3367   `'ccnv 4287   dom cdm 4288   "cima 4291    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-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-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  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-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853
This theorem is referenced by: (None)
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