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Theorem f1ocan1fv 37107
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 6827 . . . 4 (𝐺:𝐴1-1-onto𝐵𝐺:𝐴𝐵)
213ad2ant2 1131 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝐺:𝐴𝐵)
3 f1ocnv 6839 . . . . . 6 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵1-1-onto𝐴)
4 f1of 6827 . . . . . 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, 7ffvelcdmd 7081 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺𝑋) ∈ 𝐴)
9 fvco3 6984 . . 3 ((𝐺:𝐴𝐵 ∧ (𝐺𝑋) ∈ 𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹‘(𝐺‘(𝐺𝑋))))
102, 8, 9syl2anc 583 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹‘(𝐺‘(𝐺𝑋))))
11 f1ocnvfv2 7271 . . . 4 ((𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺‘(𝐺𝑋)) = 𝑋)
12113adant1 1127 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺‘(𝐺𝑋)) = 𝑋)
1312fveq2d 6889 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐹‘(𝐺‘(𝐺𝑋))) = (𝐹𝑋))
1410, 13eqtrd 2766 1 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  ccnv 5668  ccom 5673  Fun wfun 6531  wf 6533  1-1-ontowf1o 6536  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545
This theorem is referenced by:  f1ocan2fv  37108
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