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Theorem fnresdisj 4952
Description: A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
fnresdisj  F  Fn  i^i  (/)  F  |`  (/)

Proof of Theorem fnresdisj
StepHypRef Expression
1 relres 4582 . . 3  Rel  F  |`
2 reldm0 4496 . . 3  Rel  F  |`  F  |`  (/)  dom  F  |`  (/)
31, 2ax-mp 7 . 2  F  |`  (/)  dom  F  |`  (/)
4 dmres 4575 . . . . 5  dom  F  |`  i^i  dom  F
5 incom 3123 . . . . 5  i^i  dom  F  dom  F  i^i
64, 5eqtri 2057 . . . 4  dom  F  |`  dom  F  i^i
7 fndm 4941 . . . . 5  F  Fn  dom  F
87ineq1d 3131 . . . 4  F  Fn  dom  F  i^i  i^i
96, 8syl5eq 2081 . . 3  F  Fn  dom  F  |`  i^i
109eqeq1d 2045 . 2  F  Fn  dom  F  |`  (/)  i^i  (/)
113, 10syl5rbb 182 1  F  Fn  i^i  (/)  F  |`  (/)
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242    i^i cin 2910   (/)c0 3218   dom cdm 4288    |` cres 4290   Rel wrel 4293    Fn wfn 4840
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-in1 544  ax-in2 545  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-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-res 4300  df-fn 4848
This theorem is referenced by:  fvsnun2  5304  fseq1p1m1  8726
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