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Theorem nff1o 6102
 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 5864 . 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 6066 . . 3 𝑥 𝐹:𝐴1-1𝐵
62, 3, 4nffo 6081 . . 3 𝑥 𝐹:𝐴onto𝐵
75, 6nfan 1825 . 2 𝑥(𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵)
81, 7nfxfr 1776 1 𝑥 𝐹:𝐴1-1-onto𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384  Ⅎwnf 1705  Ⅎwnfc 2748  –1-1→wf1 5854  –onto→wfo 5855  –1-1-onto→wf1o 5856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864 This theorem is referenced by:  nfiso  6537  nfsum1  14370  nfsum  14371  nfcprod1  14584  nfcprod  14585  esumiun  29979  wessf1ornlem  38880  stoweidlem35  39589
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