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Theorem elmapfun 8147
 Description: A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
Assertion
Ref Expression
elmapfun (𝐴 ∈ (𝐵𝑚 𝐶) → Fun 𝐴)

Proof of Theorem elmapfun
StepHypRef Expression
1 elmapi 8145 . 2 (𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴:𝐶𝐵)
21ffund 6283 1 (𝐴 ∈ (𝐵𝑚 𝐶) → Fun 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2166  Fun wfun 6118  (class class class)co 6906   ↑𝑚 cmap 8123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-map 8125 This theorem is referenced by:  fsfnn0gsumfsffz  18733  ltbwe  19834  frlmbas  20463  islindf4  20545  mbfmfun  30862  eulerpartgbij  30980  uncf  33932  pwfi2f1o  38510  hoicvr  41557  ovnovollem1  41665  ovnovollem2  41666  domnmsuppn0  42998  rmsuppss  42999  mndpsuppss  43000  scmsuppss  43001  lincext2  43092
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