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Theorem f0cli 6025
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 6013 . 2  |-  ( C  e.  A  ->  ( F `  C )  e.  B )
31fdmi 5729 . . . 4  |-  dom  F  =  A
43eleq2i 2540 . . 3  |-  ( C  e.  dom  F  <->  C  e.  A )
5 ndmfv 5883 . . . 4  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  =  (/) )
6 f0cl.2 . . . 4  |-  (/)  e.  B
75, 6syl6eqel 2558 . . 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 1762   (/)c0 3780   dom cdm 4994   -->wf 5577   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589
This theorem is referenced by:  harcl  7978  cantnfvalf  8075  rankon  8204  cardon  8316  alephon  8441  ackbij1lem13  8603  ackbij1b  8610  ixxssxr  11532  sadcf  13953  smupf  13978  iccordt  19476
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