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Theorem f0cli 7130
Description: Unconditional closure of a function when the codomain includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)
Hypotheses
Ref Expression
f0cl.1 𝐹:𝐴𝐵
f0cl.2 ∅ ∈ 𝐵
Assertion
Ref Expression
f0cli (𝐹𝐶) ∈ 𝐵

Proof of Theorem f0cli
StepHypRef Expression
1 f0cl.1 . . 3 𝐹:𝐴𝐵
21ffvelcdmi 7115 . 2 (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
31fdmi 6757 . . . 4 dom 𝐹 = 𝐴
43eleq2i 2830 . . 3 (𝐶 ∈ dom 𝐹𝐶𝐴)
5 ndmfv 6954 . . . 4 𝐶 ∈ dom 𝐹 → (𝐹𝐶) = ∅)
6 f0cl.2 . . . 4 ∅ ∈ 𝐵
75, 6eqeltrdi 2846 . . 3 𝐶 ∈ dom 𝐹 → (𝐹𝐶) ∈ 𝐵)
84, 7sylnbir 331 . 2 𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
92, 8pm2.61i 182 1 (𝐹𝐶) ∈ 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2103  c0 4347  dom cdm 5699  wf 6568  cfv 6572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fv 6580
This theorem is referenced by:  harcl  9624  cantnfvalf  9730  rankon  9860  cardon  10009  alephon  10134  ackbij1lem13  10296  ackbij1b  10303  ixxssxr  13415  sadcf  16493  smupf  16518  iccordt  23236  nodense  27746  bdayelon  27830  madessno  27908  oldssno  27909  newssno  27910
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