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Theorem f0cli 7134
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 7119 . 2 (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
31fdmi 6760 . . . 4 dom 𝐹 = 𝐴
43eleq2i 2836 . . 3 (𝐶 ∈ dom 𝐹𝐶𝐴)
5 ndmfv 6957 . . . 4 𝐶 ∈ dom 𝐹 → (𝐹𝐶) = ∅)
6 f0cl.2 . . . 4 ∅ ∈ 𝐵
75, 6eqeltrdi 2852 . . 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 2108  c0 4352  dom cdm 5700  wf 6571  cfv 6575
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fv 6583
This theorem is referenced by:  harcl  9630  cantnfvalf  9736  rankon  9866  cardon  10015  alephon  10140  ackbij1lem13  10302  ackbij1b  10309  ixxssxr  13421  sadcf  16501  smupf  16526  iccordt  23245  nodense  27757  bdayelon  27841  madessno  27919  oldssno  27920  newssno  27921
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