Theorem fnmpti 5356

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
fnmpti.1	|- B e. _V
fnmpti.2	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
fnmpti	|- F Fn A

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem fnmpti

Step	Hyp	Ref	Expression
1		fnmpti.1	. . 3 |- B e. _V
2	1	rgenw 2542	. 2 |- A. x e. A B e. _V
3		fnmpti.2	. . 3 |- F = ( x e. A |-> B )
4	3	mptfng 5353	. 2 |- ( A. x e. A B e. _V <-> F Fn A )
5	2, 4	mpbi 145	1 |- F Fn A

Syntax hints:   = wceq 1363   e. wcel 2158   A.wral 2465   _Vcvv 2749   |-> cmpt 4076   Fn wfn 5223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-fun 5230  df-fn 5231

This theorem is referenced by:  dmmpti  5357  fconst  5423  eufnfv  5760  idref  5770  fo1st  6172  fo2nd  6173  reldm  6201  oafnex  6459  fnoei  6467  oeiexg  6468  mapsnf1o2  6710  1arith  12379  slotslfn  12502  topnfn  12711  fn0g  12813  fnmgp  13174  rlmfn  13642  fncld  13894  xmetunirn  14154