Step |
Hyp |
Ref |
Expression |
1 |
|
fsneqrn.a |
|- ( ph -> A e. V ) |
2 |
|
fsneqrn.b |
|- B = { A } |
3 |
|
fsneqrn.f |
|- ( ph -> F Fn B ) |
4 |
|
fsneqrn.g |
|- ( ph -> G Fn B ) |
5 |
|
dffn3 |
|- ( F Fn B <-> F : B --> ran F ) |
6 |
3 5
|
sylib |
|- ( ph -> F : B --> ran F ) |
7 |
|
snidg |
|- ( A e. V -> A e. { A } ) |
8 |
1 7
|
syl |
|- ( ph -> A e. { A } ) |
9 |
2
|
a1i |
|- ( ph -> B = { A } ) |
10 |
9
|
eqcomd |
|- ( ph -> { A } = B ) |
11 |
8 10
|
eleqtrd |
|- ( ph -> A e. B ) |
12 |
6 11
|
ffvelrnd |
|- ( ph -> ( F ` A ) e. ran F ) |
13 |
12
|
adantr |
|- ( ( ph /\ F = G ) -> ( F ` A ) e. ran F ) |
14 |
|
simpr |
|- ( ( ph /\ F = G ) -> F = G ) |
15 |
14
|
rneqd |
|- ( ( ph /\ F = G ) -> ran F = ran G ) |
16 |
13 15
|
eleqtrd |
|- ( ( ph /\ F = G ) -> ( F ` A ) e. ran G ) |
17 |
16
|
ex |
|- ( ph -> ( F = G -> ( F ` A ) e. ran G ) ) |
18 |
|
simpr |
|- ( ( ph /\ ( F ` A ) e. ran G ) -> ( F ` A ) e. ran G ) |
19 |
|
dffn2 |
|- ( G Fn B <-> G : B --> _V ) |
20 |
4 19
|
sylib |
|- ( ph -> G : B --> _V ) |
21 |
9
|
feq2d |
|- ( ph -> ( G : B --> _V <-> G : { A } --> _V ) ) |
22 |
20 21
|
mpbid |
|- ( ph -> G : { A } --> _V ) |
23 |
1 22
|
rnsnf |
|- ( ph -> ran G = { ( G ` A ) } ) |
24 |
23
|
adantr |
|- ( ( ph /\ ( F ` A ) e. ran G ) -> ran G = { ( G ` A ) } ) |
25 |
18 24
|
eleqtrd |
|- ( ( ph /\ ( F ` A ) e. ran G ) -> ( F ` A ) e. { ( G ` A ) } ) |
26 |
|
elsni |
|- ( ( F ` A ) e. { ( G ` A ) } -> ( F ` A ) = ( G ` A ) ) |
27 |
25 26
|
syl |
|- ( ( ph /\ ( F ` A ) e. ran G ) -> ( F ` A ) = ( G ` A ) ) |
28 |
1
|
adantr |
|- ( ( ph /\ ( F ` A ) e. ran G ) -> A e. V ) |
29 |
3
|
adantr |
|- ( ( ph /\ ( F ` A ) e. ran G ) -> F Fn B ) |
30 |
4
|
adantr |
|- ( ( ph /\ ( F ` A ) e. ran G ) -> G Fn B ) |
31 |
28 2 29 30
|
fsneq |
|- ( ( ph /\ ( F ` A ) e. ran G ) -> ( F = G <-> ( F ` A ) = ( G ` A ) ) ) |
32 |
27 31
|
mpbird |
|- ( ( ph /\ ( F ` A ) e. ran G ) -> F = G ) |
33 |
32
|
ex |
|- ( ph -> ( ( F ` A ) e. ran G -> F = G ) ) |
34 |
17 33
|
impbid |
|- ( ph -> ( F = G <-> ( F ` A ) e. ran G ) ) |