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Theorem f0cli 6018
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 6006 . 2  |-  ( C  e.  A  ->  ( F `  C )  e.  B )
31fdmi 5718 . . . 4  |-  dom  F  =  A
43eleq2i 2532 . . 3  |-  ( C  e.  dom  F  <->  C  e.  A )
5 ndmfv 5872 . . . 4  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  =  (/) )
6 f0cl.2 . . . 4  |-  (/)  e.  B
75, 6syl6eqel 2550 . . 3  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  e.  B )
84, 7sylnbir 305 . 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 1823   (/)c0 3783   dom cdm 4988   -->wf 5566   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578
This theorem is referenced by:  harcl  7979  cantnfvalf  8075  rankon  8204  cardon  8316  alephon  8441  ackbij1lem13  8603  ackbij1b  8610  ixxssxr  11544  sadcf  14187  smupf  14212  iccordt  19882
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