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Theorem f1ocan2fv 35885
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 6719 . . . . . 6 (𝐺:𝐴1-1-onto𝐵 → Rel 𝐺)
2 dfrel2 6092 . . . . . 6 (Rel 𝐺𝐺 = 𝐺)
31, 2sylib 217 . . . . 5 (𝐺:𝐴1-1-onto𝐵𝐺 = 𝐺)
433ad2ant2 1133 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → 𝐺 = 𝐺)
54fveq1d 6776 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → (𝐺𝑋) = (𝐺𝑋))
65fveq2d 6778 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = ((𝐹𝐺)‘(𝐺𝑋)))
7 f1ocnv 6728 . . 3 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵1-1-onto𝐴)
8 f1ocan1fv 35884 . . 3 ((Fun 𝐹𝐺:𝐵1-1-onto𝐴𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
97, 8syl3an2 1163 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
106, 9eqtr3d 2780 1 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2106  ccnv 5588  ccom 5593  Rel wrel 5594  Fun wfun 6427  1-1-ontowf1o 6432  cfv 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441
This theorem is referenced by: (None)
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