Metamath Proof Explorer


Theorem abrexco

Description: Composition of two image maps C ( y ) and B ( w ) . (Contributed by NM, 27-May-2013)

Ref Expression
Hypotheses abrexco.1 B V
abrexco.2 y = B C = D
Assertion abrexco x | y z | w A z = B x = C = x | w A x = D

Proof

Step Hyp Ref Expression
1 abrexco.1 B V
2 abrexco.2 y = B C = D
3 df-rex y z | w A z = B x = C y y z | w A z = B x = C
4 vex y V
5 eqeq1 z = y z = B y = B
6 5 rexbidv z = y w A z = B w A y = B
7 4 6 elab y z | w A z = B w A y = B
8 7 anbi1i y z | w A z = B x = C w A y = B x = C
9 r19.41v w A y = B x = C w A y = B x = C
10 8 9 bitr4i y z | w A z = B x = C w A y = B x = C
11 10 exbii y y z | w A z = B x = C y w A y = B x = C
12 3 11 bitri y z | w A z = B x = C y w A y = B x = C
13 rexcom4 w A y y = B x = C y w A y = B x = C
14 12 13 bitr4i y z | w A z = B x = C w A y y = B x = C
15 2 eqeq2d y = B x = C x = D
16 1 15 ceqsexv y y = B x = C x = D
17 16 rexbii w A y y = B x = C w A x = D
18 14 17 bitri y z | w A z = B x = C w A x = D
19 18 abbii x | y z | w A z = B x = C = x | w A x = D