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Theorem nff 6732
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 6565 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 nff.1 . . . 4 𝑥𝐹
3 nff.2 . . . 4 𝑥𝐴
42, 3nffn 6667 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5963 . . . 4 𝑥ran 𝐹
6 nff.3 . . . 4 𝑥𝐵
75, 6nfss 3976 . . 3 𝑥ran 𝐹𝐵
84, 7nfan 1899 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)
91, 8nfxfr 1853 1 𝑥 𝐹:𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395  wnf 1783  wnfc 2890  wss 3951  ran crn 5686   Fn wfn 6556  wf 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  nff1  6802  dffo3f  7126  nfwrd  14581  lfgrnloop  29142  fcomptf  32668  aciunf1lem  32672  fnpreimac  32681  esumfzf  34070  esumfsup  34071  poimirlem24  37651  sdclem1  37750  nfrelp  44970  fmuldfeqlem1  45597  fnlimfvre  45689  dvnmul  45958  stoweidlem53  46068  stoweidlem54  46069  stoweidlem57  46072  sge0iunmpt  46433
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