Step |
Hyp |
Ref |
Expression |
1 |
|
rnghomval.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
2 |
|
rnghomval.2 |
⊢ 𝐻 = ( 2nd ‘ 𝑅 ) |
3 |
|
rnghomval.3 |
⊢ 𝑋 = ran 𝐺 |
4 |
|
rnghomval.4 |
⊢ 𝑈 = ( GId ‘ 𝐻 ) |
5 |
|
rnghomval.5 |
⊢ 𝐽 = ( 1st ‘ 𝑆 ) |
6 |
|
rnghomval.6 |
⊢ 𝐾 = ( 2nd ‘ 𝑆 ) |
7 |
|
rnghomval.7 |
⊢ 𝑌 = ran 𝐽 |
8 |
|
rnghomval.8 |
⊢ 𝑉 = ( GId ‘ 𝐾 ) |
9 |
1 2 3 4 5 6 7 8
|
rngohomval |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) → ( 𝑅 RngHom 𝑆 ) = { 𝑓 ∈ ( 𝑌 ↑m 𝑋 ) ∣ ( ( 𝑓 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐽 ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐾 ( 𝑓 ‘ 𝑦 ) ) ) ) } ) |
10 |
9
|
eleq2d |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) → ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ↔ 𝐹 ∈ { 𝑓 ∈ ( 𝑌 ↑m 𝑋 ) ∣ ( ( 𝑓 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐽 ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐾 ( 𝑓 ‘ 𝑦 ) ) ) ) } ) ) |
11 |
5
|
fvexi |
⊢ 𝐽 ∈ V |
12 |
11
|
rnex |
⊢ ran 𝐽 ∈ V |
13 |
7 12
|
eqeltri |
⊢ 𝑌 ∈ V |
14 |
1
|
fvexi |
⊢ 𝐺 ∈ V |
15 |
14
|
rnex |
⊢ ran 𝐺 ∈ V |
16 |
3 15
|
eqeltri |
⊢ 𝑋 ∈ V |
17 |
13 16
|
elmap |
⊢ ( 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ↔ 𝐹 : 𝑋 ⟶ 𝑌 ) |
18 |
17
|
anbi1i |
⊢ ( ( 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ∧ ( ( 𝐹 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ( ( 𝐹 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ) ) |
19 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑈 ) = ( 𝐹 ‘ 𝑈 ) ) |
20 |
19
|
eqeq1d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑈 ) = 𝑉 ↔ ( 𝐹 ‘ 𝑈 ) = 𝑉 ) ) |
21 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) ) |
22 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
23 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
24 |
22 23
|
oveq12d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) 𝐽 ( 𝑓 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ) |
25 |
21 24
|
eqeq12d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐽 ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ) ) |
26 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) ) |
27 |
22 23
|
oveq12d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) 𝐾 ( 𝑓 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) |
28 |
26 27
|
eqeq12d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐾 ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) |
29 |
25 28
|
anbi12d |
⊢ ( 𝑓 = 𝐹 → ( ( ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐽 ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐾 ( 𝑓 ‘ 𝑦 ) ) ) ↔ ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
30 |
29
|
2ralbidv |
⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐽 ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐾 ( 𝑓 ‘ 𝑦 ) ) ) ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
31 |
20 30
|
anbi12d |
⊢ ( 𝑓 = 𝐹 → ( ( ( 𝑓 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐽 ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐾 ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ ( ( 𝐹 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ) ) |
32 |
31
|
elrab |
⊢ ( 𝐹 ∈ { 𝑓 ∈ ( 𝑌 ↑m 𝑋 ) ∣ ( ( 𝑓 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐽 ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐾 ( 𝑓 ‘ 𝑦 ) ) ) ) } ↔ ( 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ∧ ( ( 𝐹 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ) ) |
33 |
|
3anass |
⊢ ( ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ( 𝐹 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ( ( 𝐹 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ) ) |
34 |
18 32 33
|
3bitr4i |
⊢ ( 𝐹 ∈ { 𝑓 ∈ ( 𝑌 ↑m 𝑋 ) ∣ ( ( 𝑓 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑓 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐽 ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) 𝐾 ( 𝑓 ‘ 𝑦 ) ) ) ) } ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ( 𝐹 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
35 |
10 34
|
bitrdi |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) → ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ( 𝐹 ‘ 𝑈 ) = 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝐹 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 𝑥 𝐻 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐾 ( 𝐹 ‘ 𝑦 ) ) ) ) ) ) |