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Theorem nff1 6776
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 6539 . 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 6704 . . 3 𝑥 𝐹:𝐴𝐵
62nfcnv 5869 . . . 4 𝑥𝐹
76nffun 6562 . . 3 𝑥Fun 𝐹
85, 7nfan 1894 . 2 𝑥(𝐹:𝐴𝐵 ∧ Fun 𝐹)
91, 8nfxfr 1847 1 𝑥 𝐹:𝐴1-1𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395  wnf 1777  wnfc 2875  ccnv 5666  Fun wfun 6528  wf 6530  1-1wf1 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539
This theorem is referenced by:  nff1o  6822  iundom2g  10532
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