Step |
Hyp |
Ref |
Expression |
1 |
|
canth4.1 |
⊢ 𝑊 = { 〈 𝑥 , 𝑟 〉 ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ) ) } |
2 |
|
canth4.2 |
⊢ 𝐵 = ∪ dom 𝑊 |
3 |
|
canth4.3 |
⊢ 𝐶 = ( ◡ ( 𝑊 ‘ 𝐵 ) “ { ( 𝐹 ‘ 𝐵 ) } ) |
4 |
|
eqid |
⊢ 𝐵 = 𝐵 |
5 |
|
eqid |
⊢ ( 𝑊 ‘ 𝐵 ) = ( 𝑊 ‘ 𝐵 ) |
6 |
4 5
|
pm3.2i |
⊢ ( 𝐵 = 𝐵 ∧ ( 𝑊 ‘ 𝐵 ) = ( 𝑊 ‘ 𝐵 ) ) |
7 |
|
simp1 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → 𝐴 ∈ 𝑉 ) |
8 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ dom card ) ) → 𝐹 : 𝐷 ⟶ 𝐴 ) |
9 |
|
simp3 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) |
10 |
9
|
sselda |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ dom card ) ) → 𝑥 ∈ 𝐷 ) |
11 |
8 10
|
ffvelrnd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ dom card ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) |
12 |
1 7 11 2
|
fpwwe |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( ( 𝐵 𝑊 ( 𝑊 ‘ 𝐵 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝐵 ) ↔ ( 𝐵 = 𝐵 ∧ ( 𝑊 ‘ 𝐵 ) = ( 𝑊 ‘ 𝐵 ) ) ) ) |
13 |
6 12
|
mpbiri |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝐵 𝑊 ( 𝑊 ‘ 𝐵 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝐵 ) ) |
14 |
13
|
simpld |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → 𝐵 𝑊 ( 𝑊 ‘ 𝐵 ) ) |
15 |
1 7
|
fpwwelem |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝐵 𝑊 ( 𝑊 ‘ 𝐵 ) ↔ ( ( 𝐵 ⊆ 𝐴 ∧ ( 𝑊 ‘ 𝐵 ) ⊆ ( 𝐵 × 𝐵 ) ) ∧ ( ( 𝑊 ‘ 𝐵 ) We 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { 𝑦 } ) ) = 𝑦 ) ) ) ) |
16 |
14 15
|
mpbid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( ( 𝐵 ⊆ 𝐴 ∧ ( 𝑊 ‘ 𝐵 ) ⊆ ( 𝐵 × 𝐵 ) ) ∧ ( ( 𝑊 ‘ 𝐵 ) We 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { 𝑦 } ) ) = 𝑦 ) ) ) |
17 |
16
|
simpld |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝐵 ⊆ 𝐴 ∧ ( 𝑊 ‘ 𝐵 ) ⊆ ( 𝐵 × 𝐵 ) ) ) |
18 |
17
|
simpld |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → 𝐵 ⊆ 𝐴 ) |
19 |
|
cnvimass |
⊢ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { ( 𝐹 ‘ 𝐵 ) } ) ⊆ dom ( 𝑊 ‘ 𝐵 ) |
20 |
3 19
|
eqsstri |
⊢ 𝐶 ⊆ dom ( 𝑊 ‘ 𝐵 ) |
21 |
17
|
simprd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝑊 ‘ 𝐵 ) ⊆ ( 𝐵 × 𝐵 ) ) |
22 |
|
dmss |
⊢ ( ( 𝑊 ‘ 𝐵 ) ⊆ ( 𝐵 × 𝐵 ) → dom ( 𝑊 ‘ 𝐵 ) ⊆ dom ( 𝐵 × 𝐵 ) ) |
23 |
21 22
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → dom ( 𝑊 ‘ 𝐵 ) ⊆ dom ( 𝐵 × 𝐵 ) ) |
24 |
|
dmxpid |
⊢ dom ( 𝐵 × 𝐵 ) = 𝐵 |
25 |
23 24
|
sseqtrdi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → dom ( 𝑊 ‘ 𝐵 ) ⊆ 𝐵 ) |
26 |
20 25
|
sstrid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → 𝐶 ⊆ 𝐵 ) |
27 |
13
|
simprd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝐵 ) |
28 |
16
|
simprd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( ( 𝑊 ‘ 𝐵 ) We 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { 𝑦 } ) ) = 𝑦 ) ) |
29 |
28
|
simpld |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝑊 ‘ 𝐵 ) We 𝐵 ) |
30 |
|
weso |
⊢ ( ( 𝑊 ‘ 𝐵 ) We 𝐵 → ( 𝑊 ‘ 𝐵 ) Or 𝐵 ) |
31 |
29 30
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝑊 ‘ 𝐵 ) Or 𝐵 ) |
32 |
|
sonr |
⊢ ( ( ( 𝑊 ‘ 𝐵 ) Or 𝐵 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝐵 ) → ¬ ( 𝐹 ‘ 𝐵 ) ( 𝑊 ‘ 𝐵 ) ( 𝐹 ‘ 𝐵 ) ) |
33 |
31 27 32
|
syl2anc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ¬ ( 𝐹 ‘ 𝐵 ) ( 𝑊 ‘ 𝐵 ) ( 𝐹 ‘ 𝐵 ) ) |
34 |
3
|
eleq2i |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ↔ ( 𝐹 ‘ 𝐵 ) ∈ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { ( 𝐹 ‘ 𝐵 ) } ) ) |
35 |
|
fvex |
⊢ ( 𝐹 ‘ 𝐵 ) ∈ V |
36 |
35
|
eliniseg |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ V → ( ( 𝐹 ‘ 𝐵 ) ∈ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { ( 𝐹 ‘ 𝐵 ) } ) ↔ ( 𝐹 ‘ 𝐵 ) ( 𝑊 ‘ 𝐵 ) ( 𝐹 ‘ 𝐵 ) ) ) |
37 |
35 36
|
ax-mp |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { ( 𝐹 ‘ 𝐵 ) } ) ↔ ( 𝐹 ‘ 𝐵 ) ( 𝑊 ‘ 𝐵 ) ( 𝐹 ‘ 𝐵 ) ) |
38 |
34 37
|
bitri |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ↔ ( 𝐹 ‘ 𝐵 ) ( 𝑊 ‘ 𝐵 ) ( 𝐹 ‘ 𝐵 ) ) |
39 |
33 38
|
sylnibr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ) |
40 |
26 27 39
|
ssnelpssd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → 𝐶 ⊊ 𝐵 ) |
41 |
|
sneq |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝐵 ) → { 𝑦 } = { ( 𝐹 ‘ 𝐵 ) } ) |
42 |
41
|
imaeq2d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝐵 ) → ( ◡ ( 𝑊 ‘ 𝐵 ) “ { 𝑦 } ) = ( ◡ ( 𝑊 ‘ 𝐵 ) “ { ( 𝐹 ‘ 𝐵 ) } ) ) |
43 |
42 3
|
eqtr4di |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝐵 ) → ( ◡ ( 𝑊 ‘ 𝐵 ) “ { 𝑦 } ) = 𝐶 ) |
44 |
43
|
fveq2d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝐵 ) → ( 𝐹 ‘ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { 𝑦 } ) ) = ( 𝐹 ‘ 𝐶 ) ) |
45 |
|
id |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝐵 ) → 𝑦 = ( 𝐹 ‘ 𝐵 ) ) |
46 |
44 45
|
eqeq12d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝐵 ) → ( ( 𝐹 ‘ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { 𝑦 } ) ) = 𝑦 ↔ ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐵 ) ) ) |
47 |
28
|
simprd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( ◡ ( 𝑊 ‘ 𝐵 ) “ { 𝑦 } ) ) = 𝑦 ) |
48 |
46 47 27
|
rspcdva |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐵 ) ) |
49 |
48
|
eqcomd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐶 ) ) |
50 |
18 40 49
|
3jca |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐷 ⟶ 𝐴 ∧ ( 𝒫 𝐴 ∩ dom card ) ⊆ 𝐷 ) → ( 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐶 ) ) ) |