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Theorem f1ocan2fv 36236
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 6791 . . . . . 6 (𝐺:𝐴1-1-onto𝐵 → Rel 𝐺)
2 dfrel2 6145 . . . . . 6 (Rel 𝐺𝐺 = 𝐺)
31, 2sylib 217 . . . . 5 (𝐺:𝐴1-1-onto𝐵𝐺 = 𝐺)
433ad2ant2 1135 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → 𝐺 = 𝐺)
54fveq1d 6848 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → (𝐺𝑋) = (𝐺𝑋))
65fveq2d 6850 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = ((𝐹𝐺)‘(𝐺𝑋)))
7 f1ocnv 6800 . . 3 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵1-1-onto𝐴)
8 f1ocan1fv 36235 . . 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 2107  ccnv 5636  ccom 5641  Rel wrel 5642  Fun wfun 6494  1-1-ontowf1o 6499  cfv 6500
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508
This theorem is referenced by: (None)
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