Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  f1opr Structured version   Unicode version

Theorem f1opr 29816
 Description: Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)
Assertion
Ref Expression
f1opr
Distinct variable groups:   ,,,,   ,,,,   ,,,,
Allowed substitution hints:   (,,,)

Proof of Theorem f1opr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 6152 . 2
2 fveq2 5864 . . . . . . . . 9
3 df-ov 6285 . . . . . . . . 9
42, 3syl6eqr 2526 . . . . . . . 8
54eqeq1d 2469 . . . . . . 7
6 eqeq1 2471 . . . . . . 7
75, 6imbi12d 320 . . . . . 6
87ralbidv 2903 . . . . 5
98ralxp 5142 . . . 4
10 fveq2 5864 . . . . . . . . 9
11 df-ov 6285 . . . . . . . . 9
1210, 11syl6eqr 2526 . . . . . . . 8
1312eqeq2d 2481 . . . . . . 7
14 eqeq2 2482 . . . . . . . 8
15 vex 3116 . . . . . . . . 9
16 vex 3116 . . . . . . . . 9
1715, 16opth 4721 . . . . . . . 8
1814, 17syl6bb 261 . . . . . . 7
1913, 18imbi12d 320 . . . . . 6
2019ralxp 5142 . . . . 5
21202ralbii 2896 . . . 4
229, 21bitri 249 . . 3
2322anbi2i 694 . 2
241, 23bitri 249 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379  wral 2814  cop 4033   cxp 4997  wf 5582  wf1 5583  cfv 5586  (class class class)co 6282 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fv 5594  df-ov 6285 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator