Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  f1ocan2fv Structured version   Visualization version   GIF version

Theorem f1ocan2fv 33854
Description: Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
f1ocan2fv ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))

Proof of Theorem f1ocan2fv
StepHypRef Expression
1 f1orel 6282 . . . . . 6 (𝐺:𝐴1-1-onto𝐵 → Rel 𝐺)
2 dfrel2 5723 . . . . . 6 (Rel 𝐺𝐺 = 𝐺)
31, 2sylib 208 . . . . 5 (𝐺:𝐴1-1-onto𝐵𝐺 = 𝐺)
433ad2ant2 1128 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → 𝐺 = 𝐺)
54fveq1d 6335 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → (𝐺𝑋) = (𝐺𝑋))
65fveq2d 6337 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = ((𝐹𝐺)‘(𝐺𝑋)))
7 f1ocnv 6291 . . 3 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵1-1-onto𝐴)
8 f1ocan1fv 33853 . . 3 ((Fun 𝐹𝐺:𝐵1-1-onto𝐴𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
97, 8syl3an2 1167 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
106, 9eqtr3d 2807 1 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  ccnv 5249  ccom 5254  Rel wrel 5255  Fun wfun 6024  1-1-ontowf1o 6029  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator