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Theorem fnmpti 5219
 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 2462 . 2 𝑥𝐴 𝐵 ∈ V
3 fnmpti.2 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptfng 5216 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
52, 4mpbi 144 1 𝐹 Fn 𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1314   ∈ wcel 1463  ∀wral 2391  Vcvv 2658   ↦ cmpt 3957   Fn wfn 5086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-fun 5093  df-fn 5094 This theorem is referenced by:  dmmpti  5220  fconst  5286  eufnfv  5614  idref  5624  fo1st  6021  fo2nd  6022  reldm  6050  oafnex  6306  fnoei  6314  oeiexg  6315  mapsnf1o2  6556  slotslfn  11891  topnfn  12031  fncld  12173  xmetunirn  12433
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