Theorem funmpt2 5827

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 5826	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		funmpt2.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	funeqi 5810	. 2 ⊢ (Fun 𝐹 ↔ Fun (𝑥 ∈ 𝐴 ↦ 𝐵))
4	1, 3	mpbir 219	1 ⊢ Fun 𝐹

Syntax hints:   = wceq 1474   ↦ cmpt 4637  Fun wfun 5784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-fun 5792

This theorem is referenced by:  cantnfp1lem1  8436  tz9.12lem2  8512  tz9.12lem3  8513  rankf  8518  cardf2  8630  fin23lem30  9025  hashf1rn  12959  hashf1rnOLD  12960  qustgpopn  21681  ustn0  21782  metuval  22112  ipasslem8  26910  xppreima2  28664  funcnvmpt  28685  gsummpt2co  28945  metidval  29095  pstmval  29100  brsiga  29407  measbasedom  29426  sseqval  29611  ballotlem7  29758  sinccvglem  30654  bj-funtopon  32060  bj-ccinftydisj  32101  bj-elccinfty  32102  bj-minftyccb  32113  comptiunov2i  36841  icccncfext  38597  stoweidlem27  38744  stirlinglem14  38804  fourierdlem70  38893  fourierdlem71  38894  hoi2toco  39321  mptcfsupp  41977  lcoc0  42027  lincresunit2  42083