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Theorem nffun 4865
Description: Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
Hypothesis
Ref Expression
nffun.1 x𝐹
Assertion
Ref Expression
nffun xFun 𝐹

Proof of Theorem nffun
StepHypRef Expression
1 df-fun 4846 . 2 (Fun 𝐹 ↔ (Rel 𝐹 (𝐹𝐹) ⊆ I ))
2 nffun.1 . . . 4 x𝐹
32nfrel 4367 . . 3 xRel 𝐹
42nfcnv 4456 . . . . 5 x𝐹
52, 4nfco 4443 . . . 4 x(𝐹𝐹)
6 nfcv 2175 . . . 4 x I
75, 6nfss 2932 . . 3 x(𝐹𝐹) ⊆ I
83, 7nfan 1454 . 2 x(Rel 𝐹 (𝐹𝐹) ⊆ I )
91, 8nfxfr 1360 1 xFun 𝐹
Colors of variables: wff set class
Syntax hints:   wa 97  wnf 1346  wnfc 2162  wss 2911   I cid 4015  ccnv 4286  ccom 4291  Rel wrel 4292  Fun wfun 4838
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-fun 4846
This theorem is referenced by:  nffn  4936  nff1  5031  fliftfun  5377
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