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Theorem pmfun 7874
Description: A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
pmfun (𝐹 ∈ (𝐴pm 𝐵) → Fun 𝐹)

Proof of Theorem pmfun
StepHypRef Expression
1 elpmi 7873 . 2 (𝐹 ∈ (𝐴pm 𝐵) → (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵))
2 ffun 6046 . . 3 (𝐹:dom 𝐹𝐴 → Fun 𝐹)
32adantr 481 . 2 ((𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐵) → Fun 𝐹)
41, 3syl 17 1 (𝐹 ∈ (𝐴pm 𝐵) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1989  wss 3572  dom cdm 5112  Fun wfun 5880  wf 5882  (class class class)co 6647  pm cpm 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-pm 7857
This theorem is referenced by:  lmbr2  21057  lmff  21099  c1lip2  23755
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