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Theorem f0cli 6023
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 6011 . 2  |-  ( C  e.  A  ->  ( F `  C )  e.  B )
31fdmi 5722 . . . 4  |-  dom  F  =  A
43eleq2i 2519 . . 3  |-  ( C  e.  dom  F  <->  C  e.  A )
5 ndmfv 5876 . . . 4  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  =  (/) )
6 f0cl.2 . . . 4  |-  (/)  e.  B
75, 6syl6eqel 2537 . . 3  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  e.  B )
84, 7sylnbir 307 . 2  |-  ( -.  C  e.  A  -> 
( F `  C
)  e.  B )
92, 8pm2.61i 164 1  |-  ( F `
 C )  e.  B
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1802   (/)c0 3767   dom cdm 4985   -->wf 5570   ` cfv 5574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582
This theorem is referenced by:  harcl  7985  cantnfvalf  8082  rankon  8211  cardon  8323  alephon  8448  ackbij1lem13  8610  ackbij1b  8617  ixxssxr  11545  sadcf  13975  smupf  14000  iccordt  19581
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