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

Theorem nff 6289
Description: Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nff.1 𝑥𝐹
nff.2 𝑥𝐴
nff.3 𝑥𝐵
Assertion
Ref Expression
nff 𝑥 𝐹:𝐴𝐵

Proof of Theorem nff
StepHypRef Expression
1 df-f 6141 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 nff.1 . . . 4 𝑥𝐹
3 nff.2 . . . 4 𝑥𝐴
42, 3nffn 6234 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5616 . . . 4 𝑥ran 𝐹
6 nff.3 . . . 4 𝑥𝐵
75, 6nfss 3814 . . 3 𝑥ran 𝐹𝐵
84, 7nfan 1946 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)
91, 8nfxfr 1897 1 𝑥 𝐹:𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 386  wnf 1827  wnfc 2919  wss 3792  ran crn 5358   Fn wfn 6132  wf 6133
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
This theorem is referenced by:  nff1  6351  nfwrd  13636  lfgrnloop  26477  fcomptf  30027  aciunf1lem  30031  fnpreimac  30040  esumfzf  30733  esumfsup  30734  poimirlem24  34064  sdclem1  34168  dffo3f  40297  fmuldfeqlem1  40732  fnlimfvre  40824  dvnmul  41096  stoweidlem53  41207  stoweidlem54  41208  stoweidlem57  41211  sge0iunmpt  41569
  Copyright terms: Public domain W3C validator