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