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Theorem f0cli 5687
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 5680 . 2  |-  ( C  e.  A  ->  ( F `  C )  e.  B )
31fdmi 5410 . . . 4  |-  dom  F  =  A
43eleq2i 2360 . . 3  |-  ( C  e.  dom  F  <->  C  e.  A )
5 ndmfv 5568 . . . 4  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  =  (/) )
6 f0cl.2 . . . 4  |-  (/)  e.  B
75, 6syl6eqel 2384 . . 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 1696   (/)c0 3468   dom cdm 4705   -->wf 5267   ` cfv 5271
This theorem is referenced by:  harcl  7291  cantnfvalf  7382  rankon  7483  cardon  7593  alephon  7712  ackbij1lem13  7874  ackbij1b  7881  ixxssxr  10684  sadcf  12660  smupf  12685  iccordt  16960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
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