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Theorem dff1o6f 25195
 Description: A one-to-one onto function in terms of function values. (Contributed by FL, 1-Jan-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
dff1o6f.1
dff1o6f.2
Assertion
Ref Expression
dff1o6f
Distinct variable group:   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem dff1o6f
StepHypRef Expression
1 df-fo 5277 . . 3
21anbi1i 676 . 2
3 df-f1o 5278 . . 3
4 ancom 437 . . 3
5 fof 5467 . . . . 5
6 dff1o6f.1 . . . . . . 7
7 dff1o6f.2 . . . . . . 7
86, 7dff13f 5800 . . . . . 6
98baib 871 . . . . 5
105, 9syl 15 . . . 4
1110pm5.32i 618 . . 3
123, 4, 113bitri 262 . 2
13 df-3an 936 . 2
142, 12, 133bitr4i 268 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3a 934   wceq 1632  wnfc 2419  wral 2556   crn 4706   wfn 5266  wf 5267  wf1 5268  wfo 5269  wf1o 5270  cfv 5271 This theorem is referenced by:  trooo  25497  rltrooo  25518 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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