Step |
Hyp |
Ref |
Expression |
1 |
|
caofref.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
caofref.2 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
3 |
|
caofcom.3 |
⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝑆 ) |
4 |
|
caofass.4 |
⊢ ( 𝜑 → 𝐻 : 𝐴 ⟶ 𝑆 ) |
5 |
|
caoftrn.5 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) → 𝑥 𝑈 𝑧 ) ) |
6 |
5
|
ralrimivvva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) → 𝑥 𝑈 𝑧 ) ) |
7 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) → 𝑥 𝑈 𝑧 ) ) |
8 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) ∈ 𝑆 ) |
9 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑤 ) ∈ 𝑆 ) |
10 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐻 ‘ 𝑤 ) ∈ 𝑆 ) |
11 |
|
breq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 ) ) |
12 |
11
|
anbi1d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) ↔ ( ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) ) ) |
13 |
|
breq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( 𝑥 𝑈 𝑧 ↔ ( 𝐹 ‘ 𝑤 ) 𝑈 𝑧 ) ) |
14 |
12 13
|
imbi12d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) → 𝑥 𝑈 𝑧 ) ↔ ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) → ( 𝐹 ‘ 𝑤 ) 𝑈 𝑧 ) ) ) |
15 |
|
breq2 |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑤 ) → ( ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 ↔ ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ) ) |
16 |
|
breq1 |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑤 ) → ( 𝑦 𝑇 𝑧 ↔ ( 𝐺 ‘ 𝑤 ) 𝑇 𝑧 ) ) |
17 |
15 16
|
anbi12d |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑤 ) → ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) ↔ ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 𝑧 ) ) ) |
18 |
17
|
imbi1d |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑤 ) → ( ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) → ( 𝐹 ‘ 𝑤 ) 𝑈 𝑧 ) ↔ ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 𝑧 ) → ( 𝐹 ‘ 𝑤 ) 𝑈 𝑧 ) ) ) |
19 |
|
breq2 |
⊢ ( 𝑧 = ( 𝐻 ‘ 𝑤 ) → ( ( 𝐺 ‘ 𝑤 ) 𝑇 𝑧 ↔ ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) ) |
20 |
19
|
anbi2d |
⊢ ( 𝑧 = ( 𝐻 ‘ 𝑤 ) → ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 𝑧 ) ↔ ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) ) ) |
21 |
|
breq2 |
⊢ ( 𝑧 = ( 𝐻 ‘ 𝑤 ) → ( ( 𝐹 ‘ 𝑤 ) 𝑈 𝑧 ↔ ( 𝐹 ‘ 𝑤 ) 𝑈 ( 𝐻 ‘ 𝑤 ) ) ) |
22 |
20 21
|
imbi12d |
⊢ ( 𝑧 = ( 𝐻 ‘ 𝑤 ) → ( ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 𝑧 ) → ( 𝐹 ‘ 𝑤 ) 𝑈 𝑧 ) ↔ ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) → ( 𝐹 ‘ 𝑤 ) 𝑈 ( 𝐻 ‘ 𝑤 ) ) ) ) |
23 |
14 18 22
|
rspc3v |
⊢ ( ( ( 𝐹 ‘ 𝑤 ) ∈ 𝑆 ∧ ( 𝐺 ‘ 𝑤 ) ∈ 𝑆 ∧ ( 𝐻 ‘ 𝑤 ) ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) → 𝑥 𝑈 𝑧 ) → ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) → ( 𝐹 ‘ 𝑤 ) 𝑈 ( 𝐻 ‘ 𝑤 ) ) ) ) |
24 |
8 9 10 23
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑇 𝑧 ) → 𝑥 𝑈 𝑧 ) → ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) → ( 𝐹 ‘ 𝑤 ) 𝑈 ( 𝐻 ‘ 𝑤 ) ) ) ) |
25 |
7 24
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) → ( 𝐹 ‘ 𝑤 ) 𝑈 ( 𝐻 ‘ 𝑤 ) ) ) |
26 |
25
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑤 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) → ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑈 ( 𝐻 ‘ 𝑤 ) ) ) |
27 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
28 |
3
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn 𝐴 ) |
29 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
30 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑤 ) ) |
31 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) |
32 |
27 28 1 1 29 30 31
|
ofrfval |
⊢ ( 𝜑 → ( 𝐹 ∘r 𝑅 𝐺 ↔ ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ) ) |
33 |
4
|
ffnd |
⊢ ( 𝜑 → 𝐻 Fn 𝐴 ) |
34 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐻 ‘ 𝑤 ) = ( 𝐻 ‘ 𝑤 ) ) |
35 |
28 33 1 1 29 31 34
|
ofrfval |
⊢ ( 𝜑 → ( 𝐺 ∘r 𝑇 𝐻 ↔ ∀ 𝑤 ∈ 𝐴 ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) ) |
36 |
32 35
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝐹 ∘r 𝑅 𝐺 ∧ 𝐺 ∘r 𝑇 𝐻 ) ↔ ( ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) ) ) |
37 |
|
r19.26 |
⊢ ( ∀ 𝑤 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) ↔ ( ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) ) |
38 |
36 37
|
bitr4di |
⊢ ( 𝜑 → ( ( 𝐹 ∘r 𝑅 𝐺 ∧ 𝐺 ∘r 𝑇 𝐻 ) ↔ ∀ 𝑤 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ∧ ( 𝐺 ‘ 𝑤 ) 𝑇 ( 𝐻 ‘ 𝑤 ) ) ) ) |
39 |
27 33 1 1 29 30 34
|
ofrfval |
⊢ ( 𝜑 → ( 𝐹 ∘r 𝑈 𝐻 ↔ ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑈 ( 𝐻 ‘ 𝑤 ) ) ) |
40 |
26 38 39
|
3imtr4d |
⊢ ( 𝜑 → ( ( 𝐹 ∘r 𝑅 𝐺 ∧ 𝐺 ∘r 𝑇 𝐻 ) → 𝐹 ∘r 𝑈 𝐻 ) ) |