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Theorem nff 6202
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 6053 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 nff.1 . . . 4 𝑥𝐹
3 nff.2 . . . 4 𝑥𝐴
42, 3nffn 6148 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5523 . . . 4 𝑥ran 𝐹
6 nff.3 . . . 4 𝑥𝐵
75, 6nfss 3737 . . 3 𝑥ran 𝐹𝐵
84, 7nfan 1977 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)
91, 8nfxfr 1928 1 𝑥 𝐹:𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 383  wnf 1857  wnfc 2889  wss 3715  ran crn 5267   Fn wfn 6044  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-fun 6051  df-fn 6052  df-f 6053
This theorem is referenced by:  nff1  6260  nfwrd  13539  lfgrnloop  26240  fcomptf  29788  aciunf1lem  29792  esumfzf  30461  esumfsup  30462  poimirlem24  33764  sdclem1  33870  dffo3f  39881  fmuldfeqlem1  40335  fnlimfvre  40427  dvnmul  40679  stoweidlem53  40791  stoweidlem54  40792  stoweidlem57  40795  sge0iunmpt  41156
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