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Theorem nff1o 5152
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 4937 . 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 5118 . . 3 𝑥 𝐹:𝐴1-1𝐵
62, 3, 4nffo 5133 . . 3 𝑥 𝐹:𝐴onto𝐵
75, 6nfan 1473 . 2 𝑥(𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵)
81, 7nfxfr 1379 1 𝑥 𝐹:𝐴1-1-onto𝐵
Colors of variables: wff set class
Syntax hints:  wa 101  wnf 1365  wnfc 2181  1-1wf1 4927  ontowfo 4928  1-1-ontowf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  nfiso  5474  nfsum1  10106  nfsum  10107
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