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Theorem nffn 4995
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 4905 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 nffn.1 . . . 4 𝑥𝐹
32nffun 4924 . . 3 𝑥Fun 𝐹
42nfdm 4578 . . . 4 𝑥dom 𝐹
5 nffn.2 . . . 4 𝑥𝐴
64, 5nfeq 2185 . . 3 𝑥dom 𝐹 = 𝐴
73, 6nfan 1457 . 2 𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
81, 7nfxfr 1363 1 𝑥 𝐹 Fn 𝐴
Colors of variables: wff set class
Syntax hints:  wa 97   = wceq 1243  wnf 1349  wnfc 2165  dom cdm 4345  Fun wfun 4896   Fn wfn 4897
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-fun 4904  df-fn 4905
This theorem is referenced by:  nff  5043  nffo  5105
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