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Theorem nffo 5133
Description: Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
Hypotheses
Ref Expression
nffo.1  |-  F/_ x F
nffo.2  |-  F/_ x A
nffo.3  |-  F/_ x B
Assertion
Ref Expression
nffo  |-  F/ x  F : A -onto-> B

Proof of Theorem nffo
StepHypRef Expression
1 df-fo 4936 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
2 nffo.1 . . . 4  |-  F/_ x F
3 nffo.2 . . . 4  |-  F/_ x A
42, 3nffn 5023 . . 3  |-  F/ x  F  Fn  A
52nfrn 4607 . . . 4  |-  F/_ x ran  F
6 nffo.3 . . . 4  |-  F/_ x B
75, 6nfeq 2201 . . 3  |-  F/ x ran  F  =  B
84, 7nfan 1473 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  =  B )
91, 8nfxfr 1379 1  |-  F/ x  F : A -onto-> B
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259   F/wnf 1365   F/_wnfc 2181   ran crn 4374    Fn wfn 4925   -onto->wfo 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-fo 4936
This theorem is referenced by:  nff1o  5152
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