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Theorem f0cli 5671
Description: Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)
Hypotheses
Ref Expression
f0cl.1  |-  F : A
--> B
f0cl.2  |-  (/)  e.  B
Assertion
Ref Expression
f0cli  |-  ( F `
 C )  e.  B

Proof of Theorem f0cli
StepHypRef Expression
1 f0cl.1 . . 3  |-  F : A
--> B
21ffvelrni 5664 . 2  |-  ( C  e.  A  ->  ( F `  C )  e.  B )
31fdmi 5394 . . . 4  |-  dom  F  =  A
43eleq2i 2347 . . 3  |-  ( C  e.  dom  F  <->  C  e.  A )
5 ndmfv 5552 . . . 4  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  =  (/) )
6 f0cl.2 . . . 4  |-  (/)  e.  B
75, 6syl6eqel 2371 . . 3  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  e.  B )
84, 7sylnbir 298 . 2  |-  ( -.  C  e.  A  -> 
( F `  C
)  e.  B )
92, 8pm2.61i 156 1  |-  ( F `
 C )  e.  B
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1684   (/)c0 3455   dom cdm 4689   -->wf 5251   ` cfv 5255
This theorem is referenced by:  harcl  7275  cantnfvalf  7366  rankon  7467  cardon  7577  alephon  7696  ackbij1lem13  7858  ackbij1b  7865  ixxssxr  10668  sadcf  12644  smupf  12669  iccordt  16944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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