Theorem fnmpti 5451

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 ∈ V
fnmpti.2	⊢ 𝐹 = (𝑥 ∈ 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 ∈ V
2	1	rgenw 2585	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V
3		fnmpti.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	mptfng 5448	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
5	2, 4	mpbi 145	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ↦ cmpt 4144   Fn wfn 5312

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-fun 5319  df-fn 5320

This theorem is referenced by:  dmmpti  5452  fconst  5520  eufnfv  5869  idref  5879  fo1st  6301  fo2nd  6302  reldm  6330  oafnex  6588  fnoei  6596  oeiexg  6597  mapsnf1o2  6841  nninfctlemfo  12556  1arith  12885  slotslfn  13053  topnfn  13272  fn0g  13403  fnmgp  13880  rlmfn  14411  blfn  14509  fncld  14766  xmetunirn  15026  nnnninfex  16347  nninfnfiinf  16348