Theorem funmpt2 5927

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 5926	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		funmpt2.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	funeqi 5909	. 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵))
4	1, 3	mpbir 221	1 ⊢ Fun 𝐹

Syntax hints:   = wceq 1483   ↦ cmpt 4729  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  cantnfp1lem1  8575  tz9.12lem2  8651  tz9.12lem3  8652  rankf  8657  cardf2  8769  fin23lem30  9164  hashf1rn  13142  hashf1rnOLD  13143  funtopon  20725  qustgpopn  21923  ustn0  22024  metuval  22354  ipasslem8  27692  xppreima2  29450  funcnvmpt  29468  gsummpt2co  29780  metidval  29933  pstmval  29938  brsiga  30246  measbasedom  30265  sseqval  30450  ballotlem7  30597  sinccvglem  31566  bj-evalfun  33025  bj-ccinftydisj  33100  bj-elccinfty  33101  bj-minftyccb  33112  comptiunov2i  37998  icccncfext  40100  stoweidlem27  40244  stirlinglem14  40304  fourierdlem70  40393  fourierdlem71  40394  hoi2toco  40821  mptcfsupp  42161  lcoc0  42211  lincresunit2  42267