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Theorem nff1o 5321
Description: Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
Hypotheses
Ref Expression
nff1o.1 𝑥𝐹
nff1o.2 𝑥𝐴
nff1o.3 𝑥𝐵
Assertion
Ref Expression
nff1o 𝑥 𝐹:𝐴1-1-onto𝐵

Proof of Theorem nff1o
StepHypRef Expression
1 df-f1o 5088 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 nff1o.1 . . . 4 𝑥𝐹
3 nff1o.2 . . . 4 𝑥𝐴
4 nff1o.3 . . . 4 𝑥𝐵
52, 3, 4nff1 5284 . . 3 𝑥 𝐹:𝐴1-1𝐵
62, 3, 4nffo 5302 . . 3 𝑥 𝐹:𝐴onto𝐵
75, 6nfan 1527 . 2 𝑥(𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵)
81, 7nfxfr 1433 1 𝑥 𝐹:𝐴1-1-onto𝐵
Colors of variables: wff set class
Syntax hints:  wa 103  wnf 1419  wnfc 2242  1-1wf1 5078  ontowfo 5079  1-1-ontowf1o 5080
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-rn 4510  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088
This theorem is referenced by:  nfiso  5661  nfsum1  11017  nfsum  11018
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