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Theorem f1ocan1fv 35113
Description: Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
f1ocan1fv ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))

Proof of Theorem f1ocan1fv
StepHypRef Expression
1 f1of 6606 . . . 4 (𝐺:𝐴1-1-onto𝐵𝐺:𝐴𝐵)
213ad2ant2 1131 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝐺:𝐴𝐵)
3 f1ocnv 6618 . . . . . 6 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵1-1-onto𝐴)
4 f1of 6606 . . . . . 6 (𝐺:𝐵1-1-onto𝐴𝐺:𝐵𝐴)
53, 4syl 17 . . . . 5 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵𝐴)
653ad2ant2 1131 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝐺:𝐵𝐴)
7 simp3 1135 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝑋𝐵)
86, 7ffvelrnd 6843 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺𝑋) ∈ 𝐴)
9 fvco3 6751 . . 3 ((𝐺:𝐴𝐵 ∧ (𝐺𝑋) ∈ 𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹‘(𝐺‘(𝐺𝑋))))
102, 8, 9syl2anc 587 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹‘(𝐺‘(𝐺𝑋))))
11 f1ocnvfv2 7026 . . . 4 ((𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺‘(𝐺𝑋)) = 𝑋)
12113adant1 1127 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺‘(𝐺𝑋)) = 𝑋)
1312fveq2d 6665 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐹‘(𝐺‘(𝐺𝑋))) = (𝐹𝑋))
1410, 13eqtrd 2859 1 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  ccnv 5541  ccom 5546  Fun wfun 6337  wf 6339  1-1-ontowf1o 6342  cfv 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351
This theorem is referenced by:  f1ocan2fv  35114
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