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Theorem f0cli 6326
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:𝐴𝐵
f0cl.2 ∅ ∈ 𝐵
Assertion
Ref Expression
f0cli (𝐹𝐶) ∈ 𝐵

Proof of Theorem f0cli
StepHypRef Expression
1 f0cl.1 . . 3 𝐹:𝐴𝐵
21ffvelrni 6314 . 2 (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
31fdmi 6009 . . . 4 dom 𝐹 = 𝐴
43eleq2i 2690 . . 3 (𝐶 ∈ dom 𝐹𝐶𝐴)
5 ndmfv 6175 . . . 4 𝐶 ∈ dom 𝐹 → (𝐹𝐶) = ∅)
6 f0cl.2 . . . 4 ∅ ∈ 𝐵
75, 6syl6eqel 2706 . . 3 𝐶 ∈ dom 𝐹 → (𝐹𝐶) ∈ 𝐵)
84, 7sylnbir 321 . 2 𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
92, 8pm2.61i 176 1 (𝐹𝐶) ∈ 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 1987  c0 3891  dom cdm 5074  wf 5843  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855
This theorem is referenced by:  harcl  8410  cantnfvalf  8506  rankon  8602  cardon  8714  alephon  8836  ackbij1lem13  8998  ackbij1b  9005  ixxssxr  12129  sadcf  15099  smupf  15124  iccordt  20928
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