Metamath Proof Explorer


Theorem funcnv5mpt

Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017)

Ref Expression
Hypotheses funcnvmpt.0 x φ
funcnvmpt.1 _ x A
funcnvmpt.2 _ x F
funcnvmpt.3 F = x A B
funcnvmpt.4 φ x A B V
funcnv5mpt.1 x = z B = C
Assertion funcnv5mpt φ Fun F -1 x A z A x = z B C

Proof

Step Hyp Ref Expression
1 funcnvmpt.0 x φ
2 funcnvmpt.1 _ x A
3 funcnvmpt.2 _ x F
4 funcnvmpt.3 F = x A B
5 funcnvmpt.4 φ x A B V
6 funcnv5mpt.1 x = z B = C
7 1 2 3 4 5 funcnvmpt φ Fun F -1 y * x A y = B
8 nne ¬ B C B = C
9 eqvincg B V B = C y y = B y = C
10 5 9 syl φ x A B = C y y = B y = C
11 8 10 syl5bb φ x A ¬ B C y y = B y = C
12 11 imbi1d φ x A ¬ B C x = z y y = B y = C x = z
13 orcom x = z B C B C x = z
14 df-or B C x = z ¬ B C x = z
15 13 14 bitri x = z B C ¬ B C x = z
16 19.23v y y = B y = C x = z y y = B y = C x = z
17 12 15 16 3bitr4g φ x A x = z B C y y = B y = C x = z
18 17 ralbidv φ x A z A x = z B C z A y y = B y = C x = z
19 ralcom4 z A y y = B y = C x = z y z A y = B y = C x = z
20 18 19 bitrdi φ x A z A x = z B C y z A y = B y = C x = z
21 1 20 ralbida φ x A z A x = z B C x A y z A y = B y = C x = z
22 nfcv _ z A
23 nfv x y = C
24 6 eqeq2d x = z y = B y = C
25 2 22 23 24 rmo4f * x A y = B x A z A y = B y = C x = z
26 25 albii y * x A y = B y x A z A y = B y = C x = z
27 ralcom4 x A y z A y = B y = C x = z y x A z A y = B y = C x = z
28 26 27 bitr4i y * x A y = B x A y z A y = B y = C x = z
29 21 28 bitr4di φ x A z A x = z B C y * x A y = B
30 7 29 bitr4d φ Fun F -1 x A z A x = z B C