Theorem funmpt2 6088

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 6087	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		funmpt2.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	funeqi 6070	. 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵))
4	1, 3	mpbir 221	1 ⊢ Fun 𝐹

Syntax hints:   = wceq 1632   ↦ cmpt 4881  Fun wfun 6043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-fun 6051

This theorem is referenced by:  cantnfp1lem1  8750  tz9.12lem2  8826  tz9.12lem3  8827  rankf  8832  djuun  8962  cardf2  8979  fin23lem30  9376  hashf1rn  13355  funtopon  20947  qustgpopn  22144  ustn0  22245  metuval  22575  ipasslem8  28022  xppreima2  29780  funcnvmpt  29798  gsummpt2co  30110  metidval  30263  pstmval  30268  brsiga  30576  measbasedom  30595  sseqval  30780  ballotlem7  30927  sinccvglem  31894  bj-evalfun  33349  bj-ccinftydisj  33429  bj-elccinfty  33430  bj-minftyccb  33441  comptiunov2i  38518  icccncfext  40621  stoweidlem27  40765  stirlinglem14  40825  fourierdlem70  40914  fourierdlem71  40915  hoi2toco  41345  mptcfsupp  42689  lcoc0  42739  lincresunit2  42795