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Theorem nffn 4921
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 4832 . 2 (𝐹 Fn A ↔ (Fun 𝐹 dom 𝐹 = A))
2 nffn.1 . . . 4 x𝐹
32nffun 4850 . . 3 xFun 𝐹
42nfdm 4505 . . . 4 xdom 𝐹
5 nffn.2 . . . 4 xA
64, 5nfeq 2167 . . 3 xdom 𝐹 = A
73, 6nfan 1439 . 2 x(Fun 𝐹 dom 𝐹 = A)
81, 7nfxfr 1343 1 x 𝐹 Fn A
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wnf 1329  wnfc 2147  dom cdm 4272  Fun wfun 4823   Fn wfn 4824
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-fun 4831  df-fn 4832
This theorem is referenced by:  nff  4969  nffo  5030
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