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Theorem nff1o 6294
 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 6054 . 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 6258 . . 3 𝑥 𝐹:𝐴1-1𝐵
62, 3, 4nffo 6273 . . 3 𝑥 𝐹:𝐴onto𝐵
75, 6nfan 1975 . 2 𝑥(𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵)
81, 7nfxfr 1926 1 𝑥 𝐹:𝐴1-1-onto𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383  Ⅎwnf 1855  Ⅎwnfc 2887  –1-1→wf1 6044  –onto→wfo 6045  –1-1-onto→wf1o 6046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054 This theorem is referenced by:  nfiso  6733  nfsum1  14617  nfsum  14618  nfcprod1  14837  nfcprod  14838  fsumiunle  29882  esumiun  30463  wessf1ornlem  39868  stoweidlem35  40753
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