Step |
Hyp |
Ref |
Expression |
1 |
|
wemapso.t |
⊢ 𝑇 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑧 ∈ 𝐴 ( ( 𝑥 ‘ 𝑧 ) 𝑆 ( 𝑦 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ) ) } |
2 |
|
wemapsolem.1 |
⊢ 𝑈 ⊆ ( 𝐵 ↑m 𝐴 ) |
3 |
|
wemapsolem.2 |
⊢ ( 𝜑 → 𝑅 Or 𝐴 ) |
4 |
|
wemapsolem.3 |
⊢ ( 𝜑 → 𝑆 Or 𝐵 ) |
5 |
|
wemapsolem.4 |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ∃ 𝑐 ∈ dom ( 𝑎 ∖ 𝑏 ) ∀ 𝑑 ∈ dom ( 𝑎 ∖ 𝑏 ) ¬ 𝑑 𝑅 𝑐 ) |
6 |
|
sopo |
⊢ ( 𝑆 Or 𝐵 → 𝑆 Po 𝐵 ) |
7 |
4 6
|
syl |
⊢ ( 𝜑 → 𝑆 Po 𝐵 ) |
8 |
1
|
wemappo |
⊢ ( ( 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵 ) → 𝑇 Po ( 𝐵 ↑m 𝐴 ) ) |
9 |
3 7 8
|
syl2anc |
⊢ ( 𝜑 → 𝑇 Po ( 𝐵 ↑m 𝐴 ) ) |
10 |
|
poss |
⊢ ( 𝑈 ⊆ ( 𝐵 ↑m 𝐴 ) → ( 𝑇 Po ( 𝐵 ↑m 𝐴 ) → 𝑇 Po 𝑈 ) ) |
11 |
2 9 10
|
mpsyl |
⊢ ( 𝜑 → 𝑇 Po 𝑈 ) |
12 |
|
df-ne |
⊢ ( 𝑎 ≠ 𝑏 ↔ ¬ 𝑎 = 𝑏 ) |
13 |
|
simprll |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → 𝑎 ∈ 𝑈 ) |
14 |
2 13
|
sselid |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → 𝑎 ∈ ( 𝐵 ↑m 𝐴 ) ) |
15 |
|
elmapi |
⊢ ( 𝑎 ∈ ( 𝐵 ↑m 𝐴 ) → 𝑎 : 𝐴 ⟶ 𝐵 ) |
16 |
14 15
|
syl |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → 𝑎 : 𝐴 ⟶ 𝐵 ) |
17 |
16
|
ffnd |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → 𝑎 Fn 𝐴 ) |
18 |
|
simprlr |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → 𝑏 ∈ 𝑈 ) |
19 |
2 18
|
sselid |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → 𝑏 ∈ ( 𝐵 ↑m 𝐴 ) ) |
20 |
|
elmapi |
⊢ ( 𝑏 ∈ ( 𝐵 ↑m 𝐴 ) → 𝑏 : 𝐴 ⟶ 𝐵 ) |
21 |
19 20
|
syl |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → 𝑏 : 𝐴 ⟶ 𝐵 ) |
22 |
21
|
ffnd |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → 𝑏 Fn 𝐴 ) |
23 |
|
fndmdif |
⊢ ( ( 𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴 ) → dom ( 𝑎 ∖ 𝑏 ) = { 𝑥 ∈ 𝐴 ∣ ( 𝑎 ‘ 𝑥 ) ≠ ( 𝑏 ‘ 𝑥 ) } ) |
24 |
17 22 23
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → dom ( 𝑎 ∖ 𝑏 ) = { 𝑥 ∈ 𝐴 ∣ ( 𝑎 ‘ 𝑥 ) ≠ ( 𝑏 ‘ 𝑥 ) } ) |
25 |
24
|
eleq2d |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( 𝑐 ∈ dom ( 𝑎 ∖ 𝑏 ) ↔ 𝑐 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝑎 ‘ 𝑥 ) ≠ ( 𝑏 ‘ 𝑥 ) } ) ) |
26 |
|
nesym |
⊢ ( ( 𝑎 ‘ 𝑥 ) ≠ ( 𝑏 ‘ 𝑥 ) ↔ ¬ ( 𝑏 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑥 ) ) |
27 |
|
fveq2 |
⊢ ( 𝑥 = 𝑐 → ( 𝑏 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑐 ) ) |
28 |
|
fveq2 |
⊢ ( 𝑥 = 𝑐 → ( 𝑎 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑐 ) ) |
29 |
27 28
|
eqeq12d |
⊢ ( 𝑥 = 𝑐 → ( ( 𝑏 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑥 ) ↔ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ) ) |
30 |
29
|
notbid |
⊢ ( 𝑥 = 𝑐 → ( ¬ ( 𝑏 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑥 ) ↔ ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ) ) |
31 |
26 30
|
syl5bb |
⊢ ( 𝑥 = 𝑐 → ( ( 𝑎 ‘ 𝑥 ) ≠ ( 𝑏 ‘ 𝑥 ) ↔ ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ) ) |
32 |
31
|
elrab |
⊢ ( 𝑐 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝑎 ‘ 𝑥 ) ≠ ( 𝑏 ‘ 𝑥 ) } ↔ ( 𝑐 ∈ 𝐴 ∧ ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ) ) |
33 |
25 32
|
bitrdi |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( 𝑐 ∈ dom ( 𝑎 ∖ 𝑏 ) ↔ ( 𝑐 ∈ 𝐴 ∧ ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ) ) ) |
34 |
24
|
eleq2d |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( 𝑑 ∈ dom ( 𝑎 ∖ 𝑏 ) ↔ 𝑑 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝑎 ‘ 𝑥 ) ≠ ( 𝑏 ‘ 𝑥 ) } ) ) |
35 |
|
fveq2 |
⊢ ( 𝑥 = 𝑑 → ( 𝑏 ‘ 𝑥 ) = ( 𝑏 ‘ 𝑑 ) ) |
36 |
|
fveq2 |
⊢ ( 𝑥 = 𝑑 → ( 𝑎 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑑 ) ) |
37 |
35 36
|
eqeq12d |
⊢ ( 𝑥 = 𝑑 → ( ( 𝑏 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑥 ) ↔ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) |
38 |
37
|
notbid |
⊢ ( 𝑥 = 𝑑 → ( ¬ ( 𝑏 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑥 ) ↔ ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) |
39 |
26 38
|
syl5bb |
⊢ ( 𝑥 = 𝑑 → ( ( 𝑎 ‘ 𝑥 ) ≠ ( 𝑏 ‘ 𝑥 ) ↔ ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) |
40 |
39
|
elrab |
⊢ ( 𝑑 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝑎 ‘ 𝑥 ) ≠ ( 𝑏 ‘ 𝑥 ) } ↔ ( 𝑑 ∈ 𝐴 ∧ ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) |
41 |
34 40
|
bitrdi |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( 𝑑 ∈ dom ( 𝑎 ∖ 𝑏 ) ↔ ( 𝑑 ∈ 𝐴 ∧ ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) |
42 |
41
|
imbi1d |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( ( 𝑑 ∈ dom ( 𝑎 ∖ 𝑏 ) → ¬ 𝑑 𝑅 𝑐 ) ↔ ( ( 𝑑 ∈ 𝐴 ∧ ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) → ¬ 𝑑 𝑅 𝑐 ) ) ) |
43 |
|
impexp |
⊢ ( ( ( 𝑑 ∈ 𝐴 ∧ ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) → ¬ 𝑑 𝑅 𝑐 ) ↔ ( 𝑑 ∈ 𝐴 → ( ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) → ¬ 𝑑 𝑅 𝑐 ) ) ) |
44 |
|
con34b |
⊢ ( ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ↔ ( ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) → ¬ 𝑑 𝑅 𝑐 ) ) |
45 |
44
|
imbi2i |
⊢ ( ( 𝑑 ∈ 𝐴 → ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ↔ ( 𝑑 ∈ 𝐴 → ( ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) → ¬ 𝑑 𝑅 𝑐 ) ) ) |
46 |
43 45
|
bitr4i |
⊢ ( ( ( 𝑑 ∈ 𝐴 ∧ ¬ ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) → ¬ 𝑑 𝑅 𝑐 ) ↔ ( 𝑑 ∈ 𝐴 → ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) |
47 |
42 46
|
bitrdi |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( ( 𝑑 ∈ dom ( 𝑎 ∖ 𝑏 ) → ¬ 𝑑 𝑅 𝑐 ) ↔ ( 𝑑 ∈ 𝐴 → ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ) |
48 |
47
|
ralbidv2 |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( ∀ 𝑑 ∈ dom ( 𝑎 ∖ 𝑏 ) ¬ 𝑑 𝑅 𝑐 ↔ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) |
49 |
33 48
|
anbi12d |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( ( 𝑐 ∈ dom ( 𝑎 ∖ 𝑏 ) ∧ ∀ 𝑑 ∈ dom ( 𝑎 ∖ 𝑏 ) ¬ 𝑑 𝑅 𝑐 ) ↔ ( ( 𝑐 ∈ 𝐴 ∧ ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ) |
50 |
|
anass |
⊢ ( ( ( 𝑐 ∈ 𝐴 ∧ ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ↔ ( 𝑐 ∈ 𝐴 ∧ ( ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ) |
51 |
49 50
|
bitrdi |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( ( 𝑐 ∈ dom ( 𝑎 ∖ 𝑏 ) ∧ ∀ 𝑑 ∈ dom ( 𝑎 ∖ 𝑏 ) ¬ 𝑑 𝑅 𝑐 ) ↔ ( 𝑐 ∈ 𝐴 ∧ ( ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ) ) |
52 |
51
|
rexbidv2 |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( ∃ 𝑐 ∈ dom ( 𝑎 ∖ 𝑏 ) ∀ 𝑑 ∈ dom ( 𝑎 ∖ 𝑏 ) ¬ 𝑑 𝑅 𝑐 ↔ ∃ 𝑐 ∈ 𝐴 ( ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ) |
53 |
5 52
|
mpbid |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ∃ 𝑐 ∈ 𝐴 ( ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) |
54 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) ∧ 𝑐 ∈ 𝐴 ) → 𝑆 Or 𝐵 ) |
55 |
21
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) ∧ 𝑐 ∈ 𝐴 ) → ( 𝑏 ‘ 𝑐 ) ∈ 𝐵 ) |
56 |
16
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) ∧ 𝑐 ∈ 𝐴 ) → ( 𝑎 ‘ 𝑐 ) ∈ 𝐵 ) |
57 |
|
sotrieq |
⊢ ( ( 𝑆 Or 𝐵 ∧ ( ( 𝑏 ‘ 𝑐 ) ∈ 𝐵 ∧ ( 𝑎 ‘ 𝑐 ) ∈ 𝐵 ) ) → ( ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ↔ ¬ ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ) ) |
58 |
57
|
con2bid |
⊢ ( ( 𝑆 Or 𝐵 ∧ ( ( 𝑏 ‘ 𝑐 ) ∈ 𝐵 ∧ ( 𝑎 ‘ 𝑐 ) ∈ 𝐵 ) ) → ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ↔ ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ) ) |
59 |
58
|
biimprd |
⊢ ( ( 𝑆 Or 𝐵 ∧ ( ( 𝑏 ‘ 𝑐 ) ∈ 𝐵 ∧ ( 𝑎 ‘ 𝑐 ) ∈ 𝐵 ) ) → ( ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) → ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ) ) |
60 |
54 55 56 59
|
syl12anc |
⊢ ( ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) ∧ 𝑐 ∈ 𝐴 ) → ( ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) → ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ) ) |
61 |
60
|
anim1d |
⊢ ( ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) ∧ 𝑐 ∈ 𝐴 ) → ( ( ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) → ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ) |
62 |
61
|
reximdva |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( ∃ 𝑐 ∈ 𝐴 ( ¬ ( 𝑏 ‘ 𝑐 ) = ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) → ∃ 𝑐 ∈ 𝐴 ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ) |
63 |
53 62
|
mpd |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ∃ 𝑐 ∈ 𝐴 ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) |
64 |
1
|
wemaplem1 |
⊢ ( ( 𝑏 ∈ V ∧ 𝑎 ∈ V ) → ( 𝑏 𝑇 𝑎 ↔ ∃ 𝑐 ∈ 𝐴 ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ) |
65 |
64
|
el2v |
⊢ ( 𝑏 𝑇 𝑎 ↔ ∃ 𝑐 ∈ 𝐴 ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) |
66 |
1
|
wemaplem1 |
⊢ ( ( 𝑎 ∈ V ∧ 𝑏 ∈ V ) → ( 𝑎 𝑇 𝑏 ↔ ∃ 𝑐 ∈ 𝐴 ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) ) ) |
67 |
66
|
el2v |
⊢ ( 𝑎 𝑇 𝑏 ↔ ∃ 𝑐 ∈ 𝐴 ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) ) |
68 |
65 67
|
orbi12i |
⊢ ( ( 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ↔ ( ∃ 𝑐 ∈ 𝐴 ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ∨ ∃ 𝑐 ∈ 𝐴 ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) ) ) |
69 |
|
r19.43 |
⊢ ( ∃ 𝑐 ∈ 𝐴 ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ∨ ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) ) ↔ ( ∃ 𝑐 ∈ 𝐴 ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ∨ ∃ 𝑐 ∈ 𝐴 ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) ) ) |
70 |
|
andir |
⊢ ( ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ↔ ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ∨ ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ) |
71 |
|
eqcom |
⊢ ( ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ↔ ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) |
72 |
71
|
imbi2i |
⊢ ( ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ↔ ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) |
73 |
72
|
ralbii |
⊢ ( ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ↔ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) |
74 |
73
|
anbi2i |
⊢ ( ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ↔ ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) ) |
75 |
74
|
orbi2i |
⊢ ( ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ∨ ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) ↔ ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ∨ ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) ) ) |
76 |
70 75
|
bitr2i |
⊢ ( ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ∨ ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) ) ↔ ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) |
77 |
76
|
rexbii |
⊢ ( ∃ 𝑐 ∈ 𝐴 ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ∨ ( ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑎 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) ) ) ↔ ∃ 𝑐 ∈ 𝐴 ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) |
78 |
68 69 77
|
3bitr2i |
⊢ ( ( 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ↔ ∃ 𝑐 ∈ 𝐴 ( ( ( 𝑏 ‘ 𝑐 ) 𝑆 ( 𝑎 ‘ 𝑐 ) ∨ ( 𝑎 ‘ 𝑐 ) 𝑆 ( 𝑏 ‘ 𝑐 ) ) ∧ ∀ 𝑑 ∈ 𝐴 ( 𝑑 𝑅 𝑐 → ( 𝑏 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) ) ) |
79 |
63 78
|
sylibr |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ≠ 𝑏 ) ) → ( 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ) |
80 |
79
|
expr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ) → ( 𝑎 ≠ 𝑏 → ( 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ) ) |
81 |
12 80
|
syl5bir |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ) → ( ¬ 𝑎 = 𝑏 → ( 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ) ) |
82 |
81
|
orrd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ) → ( 𝑎 = 𝑏 ∨ ( 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ) ) |
83 |
|
3orrot |
⊢ ( ( 𝑎 𝑇 𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏 𝑇 𝑎 ) ↔ ( 𝑎 = 𝑏 ∨ 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ) |
84 |
|
3orass |
⊢ ( ( 𝑎 = 𝑏 ∨ 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ↔ ( 𝑎 = 𝑏 ∨ ( 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ) ) |
85 |
83 84
|
bitr2i |
⊢ ( ( 𝑎 = 𝑏 ∨ ( 𝑏 𝑇 𝑎 ∨ 𝑎 𝑇 𝑏 ) ) ↔ ( 𝑎 𝑇 𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏 𝑇 𝑎 ) ) |
86 |
82 85
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ) → ( 𝑎 𝑇 𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏 𝑇 𝑎 ) ) |
87 |
11 86
|
issod |
⊢ ( 𝜑 → 𝑇 Or 𝑈 ) |