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Theorem nffn 4936
Description: Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
Hypotheses
Ref Expression
nffn.1 x𝐹
nffn.2 xA
Assertion
Ref Expression
nffn x 𝐹 Fn A

Proof of Theorem nffn
StepHypRef Expression
1 df-fn 4847 . 2 (𝐹 Fn A ↔ (Fun 𝐹 dom 𝐹 = A))
2 nffn.1 . . . 4 x𝐹
32nffun 4865 . . 3 xFun 𝐹
42nfdm 4520 . . . 4 xdom 𝐹
5 nffn.2 . . . 4 xA
64, 5nfeq 2182 . . 3 xdom 𝐹 = A
73, 6nfan 1454 . 2 x(Fun 𝐹 dom 𝐹 = A)
81, 7nfxfr 1360 1 x 𝐹 Fn A
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wnf 1346  wnfc 2162  dom cdm 4287  Fun wfun 4838   Fn wfn 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-fun 4846  df-fn 4847
This theorem is referenced by:  nff  4984  nffo  5046
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