MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nff1 Structured version   Visualization version   GIF version

Theorem nff1 6351
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 6142 . 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 6289 . . 3 𝑥 𝐹:𝐴𝐵
62nfcnv 5548 . . . 4 𝑥𝐹
76nffun 6160 . . 3 𝑥Fun 𝐹
85, 7nfan 1946 . 2 𝑥(𝐹:𝐴𝐵 ∧ Fun 𝐹)
91, 8nfxfr 1897 1 𝑥 𝐹:𝐴1-1𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 386  wnf 1827  wnfc 2919  ccnv 5356  Fun wfun 6131  wf 6133  1-1wf1 6134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142
This theorem is referenced by:  nff1o  6391  iundom2g  9699
  Copyright terms: Public domain W3C validator