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Theorem nffn 5177
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 5084 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 nffn.1 . . . 4 𝑥𝐹
32nffun 5104 . . 3 𝑥Fun 𝐹
42nfdm 4743 . . . 4 𝑥dom 𝐹
5 nffn.2 . . . 4 𝑥𝐴
64, 5nfeq 2263 . . 3 𝑥dom 𝐹 = 𝐴
73, 6nfan 1527 . 2 𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
81, 7nfxfr 1433 1 𝑥 𝐹 Fn 𝐴
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wnf 1419  wnfc 2242  dom cdm 4499  Fun wfun 5075   Fn wfn 5076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-fun 5083  df-fn 5084
This theorem is referenced by:  nff  5227  nffo  5302  nfixpxy  6565  nfixp1  6566
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