Theorem funmpt2 5965

Description: Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)

Hypothesis

Ref	Expression
funmpt2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
funmpt2	⊢ Fun 𝐹

Proof of Theorem funmpt2

Step	Hyp	Ref	Expression
1		funmpt 5964	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		funmpt2.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	funeqi 5947	. 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵))
4	1, 3	mpbir 221	1 ⊢ Fun 𝐹

Syntax hints:   = wceq 1523   ↦ cmpt 4762  Fun wfun 5920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-fun 5928

This theorem is referenced by:  cantnfp1lem1  8613  tz9.12lem2  8689  tz9.12lem3  8690  rankf  8695  cardf2  8807  fin23lem30  9202  hashf1rn  13181  funtopon  20773  qustgpopn  21970  ustn0  22071  metuval  22401  ipasslem8  27820  xppreima2  29578  funcnvmpt  29596  gsummpt2co  29908  metidval  30061  pstmval  30066  brsiga  30374  measbasedom  30393  sseqval  30578  ballotlem7  30725  sinccvglem  31692  bj-evalfun  33150  bj-ccinftydisj  33230  bj-elccinfty  33231  bj-minftyccb  33242  comptiunov2i  38315  icccncfext  40418  stoweidlem27  40562  stirlinglem14  40622  fourierdlem70  40711  fourierdlem71  40712  hoi2toco  41142  mptcfsupp  42486  lcoc0  42536  lincresunit2  42592