Metamath Proof Explorer


Theorem f1ocan1fv

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)

Ref Expression
Assertion f1ocan1fv Fun F G : A 1-1 onto B X B F G G -1 X = F X

Proof

Step Hyp Ref Expression
1 f1of G : A 1-1 onto B G : A B
2 1 3ad2ant2 Fun F G : A 1-1 onto B X B G : A B
3 f1ocnv G : A 1-1 onto B G -1 : B 1-1 onto A
4 f1of G -1 : B 1-1 onto A G -1 : B A
5 3 4 syl G : A 1-1 onto B G -1 : B A
6 5 3ad2ant2 Fun F G : A 1-1 onto B X B G -1 : B A
7 simp3 Fun F G : A 1-1 onto B X B X B
8 6 7 ffvelrnd Fun F G : A 1-1 onto B X B G -1 X A
9 fvco3 G : A B G -1 X A F G G -1 X = F G G -1 X
10 2 8 9 syl2anc Fun F G : A 1-1 onto B X B F G G -1 X = F G G -1 X
11 f1ocnvfv2 G : A 1-1 onto B X B G G -1 X = X
12 11 3adant1 Fun F G : A 1-1 onto B X B G G -1 X = X
13 12 fveq2d Fun F G : A 1-1 onto B X B F G G -1 X = F X
14 10 13 eqtrd Fun F G : A 1-1 onto B X B F G G -1 X = F X