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 35508
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 6621 . . . . . 6 (𝐺:𝐴1-1-onto𝐵 → Rel 𝐺)
2 dfrel2 6021 . . . . . 6 (Rel 𝐺𝐺 = 𝐺)
31, 2sylib 221 . . . . 5 (𝐺:𝐴1-1-onto𝐵𝐺 = 𝐺)
433ad2ant2 1135 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → 𝐺 = 𝐺)
54fveq1d 6676 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → (𝐺𝑋) = (𝐺𝑋))
65fveq2d 6678 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = ((𝐹𝐺)‘(𝐺𝑋)))
7 f1ocnv 6630 . . 3 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵1-1-onto𝐴)
8 f1ocan1fv 35507 . . 3 ((Fun 𝐹𝐺:𝐵1-1-onto𝐴𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
97, 8syl3an2 1165 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
106, 9eqtr3d 2775 1 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2114  ccnv 5524  ccom 5529  Rel wrel 5530  Fun wfun 6333  1-1-ontowf1o 6338  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator