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Theorem fnmpti 5142
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 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fnmpti 𝐹 Fn 𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fnmpti
StepHypRef Expression
1 fnmpti.1 . . 3 𝐵 ∈ V
21rgenw 2430 . 2 𝑥𝐴 𝐵 ∈ V
3 fnmpti.2 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptfng 5139 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
52, 4mpbi 143 1 𝐹 Fn 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  wral 2359  Vcvv 2619  cmpt 3899   Fn wfn 5010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-fun 5017  df-fn 5018
This theorem is referenced by:  dmmpti  5143  fconst  5206  eufnfv  5525  idref  5536  fo1st  5928  fo2nd  5929  reldm  5956  oafnex  6205  fnoei  6213  oeiexg  6214  mapsnf1o2  6451  slotfni  11510
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