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Theorem f1ocan2fv 33493
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 6127 . . . . . 6 (𝐺:𝐴1-1-onto𝐵 → Rel 𝐺)
2 dfrel2 5571 . . . . . 6 (Rel 𝐺𝐺 = 𝐺)
31, 2sylib 208 . . . . 5 (𝐺:𝐴1-1-onto𝐵𝐺 = 𝐺)
433ad2ant2 1081 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → 𝐺 = 𝐺)
54fveq1d 6180 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → (𝐺𝑋) = (𝐺𝑋))
65fveq2d 6182 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = ((𝐹𝐺)‘(𝐺𝑋)))
7 f1ocnv 6136 . . 3 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵1-1-onto𝐴)
8 f1ocan1fv 33492 . . 3 ((Fun 𝐹𝐺:𝐵1-1-onto𝐴𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
97, 8syl3an2 1358 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
106, 9eqtr3d 2656 1 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  ccnv 5103  ccom 5108  Rel wrel 5109  Fun wfun 5870  1-1-ontowf1o 5875  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884
This theorem is referenced by: (None)
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