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Theorem imadisj 4630
Description: A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
imadisj  "  (/)  dom  i^i  (/)

Proof of Theorem imadisj
StepHypRef Expression
1 df-ima 4301 . . 3 
" 
ran  |`
21eqeq1i 2044 . 2  "  (/)  ran  |`  (/)
3 dm0rn0 4495 . 2  dom  |`  (/)  ran  |`  (/)
4 dmres 4575 . . . 4  dom  |`  i^i  dom
5 incom 3123 . . . 4  i^i  dom  dom  i^i
64, 5eqtri 2057 . . 3  dom  |`  dom  i^i
76eqeq1i 2044 . 2  dom  |`  (/)  dom  i^i  (/)
82, 3, 73bitr2i 197 1  "  (/)  dom  i^i  (/)
Colors of variables: wff set class
Syntax hints:   wb 98   wceq 1242    i^i cin 2910   (/)c0 3218   dom cdm 4288   ran crn 4289    |` cres 4290   "cima 4291
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-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-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-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  fnimadisj  4962  fnimaeq0  4963  fimacnvdisj  5017
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