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Theorem nff1 5296
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 5098 . 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 5239 . . 3 𝑥 𝐹:𝐴𝐵
62nfcnv 4688 . . . 4 𝑥𝐹
76nffun 5116 . . 3 𝑥Fun 𝐹
85, 7nfan 1529 . 2 𝑥(𝐹:𝐴𝐵 ∧ Fun 𝐹)
91, 8nfxfr 1435 1 𝑥 𝐹:𝐴1-1𝐵
Colors of variables: wff set class
Syntax hints:  wa 103  wnf 1421  wnfc 2245  ccnv 4508  Fun wfun 5087  wf 5089  1-1wf1 5090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098
This theorem is referenced by:  nff1o  5333
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