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Theorem nff1 5993
Description: Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
Hypotheses
Ref Expression
nff1.1 𝑥𝐹
nff1.2 𝑥𝐴
nff1.3 𝑥𝐵
Assertion
Ref Expression
nff1 𝑥 𝐹:𝐴1-1𝐵

Proof of Theorem nff1
StepHypRef Expression
1 df-f1 5791 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
2 nff1.1 . . . 4 𝑥𝐹
3 nff1.2 . . . 4 𝑥𝐴
4 nff1.3 . . . 4 𝑥𝐵
52, 3, 4nff 5936 . . 3 𝑥 𝐹:𝐴𝐵
62nfcnv 5207 . . . 4 𝑥𝐹
76nffun 5808 . . 3 𝑥Fun 𝐹
85, 7nfan 1814 . 2 𝑥(𝐹:𝐴𝐵 ∧ Fun 𝐹)
91, 8nfxfr 1769 1 𝑥 𝐹:𝐴1-1𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 382  wnf 1698  wnfc 2733  ccnv 5023  Fun wfun 5780  wf 5782  1-1wf1 5783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791
This theorem is referenced by:  nff1o  6029  iundom2g  9214
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