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 |
|
wemaplem2.p |
⊢ ( 𝜑 → 𝑃 ∈ ( 𝐵 ↑m 𝐴 ) ) |
3 |
|
wemaplem2.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ↑m 𝐴 ) ) |
4 |
|
wemaplem2.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝐵 ↑m 𝐴 ) ) |
5 |
|
wemaplem2.r |
⊢ ( 𝜑 → 𝑅 Or 𝐴 ) |
6 |
|
wemaplem2.s |
⊢ ( 𝜑 → 𝑆 Po 𝐵 ) |
7 |
|
wemaplem3.px |
⊢ ( 𝜑 → 𝑃 𝑇 𝑋 ) |
8 |
|
wemaplem3.xq |
⊢ ( 𝜑 → 𝑋 𝑇 𝑄 ) |
9 |
1
|
wemaplem1 |
⊢ ( ( 𝑃 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐴 ) ) → ( 𝑃 𝑇 𝑋 ↔ ∃ 𝑎 ∈ 𝐴 ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) |
10 |
2 3 9
|
syl2anc |
⊢ ( 𝜑 → ( 𝑃 𝑇 𝑋 ↔ ∃ 𝑎 ∈ 𝐴 ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) |
11 |
7 10
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑎 ∈ 𝐴 ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) |
12 |
1
|
wemaplem1 |
⊢ ( ( 𝑋 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑄 ∈ ( 𝐵 ↑m 𝐴 ) ) → ( 𝑋 𝑇 𝑄 ↔ ∃ 𝑏 ∈ 𝐴 ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) |
13 |
3 4 12
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 𝑇 𝑄 ↔ ∃ 𝑏 ∈ 𝐴 ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) |
14 |
8 13
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑏 ∈ 𝐴 ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) |
15 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → 𝑃 ∈ ( 𝐵 ↑m 𝐴 ) ) |
16 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → 𝑋 ∈ ( 𝐵 ↑m 𝐴 ) ) |
17 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → 𝑄 ∈ ( 𝐵 ↑m 𝐴 ) ) |
18 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → 𝑅 Or 𝐴 ) |
19 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → 𝑆 Po 𝐵 ) |
20 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → 𝑎 ∈ 𝐴 ) |
21 |
|
simp2rl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ) |
22 |
21
|
3expa |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ) |
23 |
|
simprr |
⊢ ( ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) → ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) |
24 |
23
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) |
25 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → 𝑏 ∈ 𝐴 ) |
26 |
|
simprrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ) |
27 |
|
simprrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) |
28 |
1 15 16 17 18 19 20 22 24 25 26 27
|
wemaplem2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) ∧ ( 𝑏 ∈ 𝐴 ∧ ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) ) ) → 𝑃 𝑇 𝑄 ) |
29 |
28
|
rexlimdvaa |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) ) ) → ( ∃ 𝑏 ∈ 𝐴 ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) → 𝑃 𝑇 𝑄 ) ) |
30 |
29
|
rexlimdvaa |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ 𝐴 ( ( 𝑃 ‘ 𝑎 ) 𝑆 ( 𝑋 ‘ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑎 → ( 𝑃 ‘ 𝑐 ) = ( 𝑋 ‘ 𝑐 ) ) ) → ( ∃ 𝑏 ∈ 𝐴 ( ( 𝑋 ‘ 𝑏 ) 𝑆 ( 𝑄 ‘ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝐴 ( 𝑐 𝑅 𝑏 → ( 𝑋 ‘ 𝑐 ) = ( 𝑄 ‘ 𝑐 ) ) ) → 𝑃 𝑇 𝑄 ) ) ) |
31 |
11 14 30
|
mp2d |
⊢ ( 𝜑 → 𝑃 𝑇 𝑄 ) |