Step |
Hyp |
Ref |
Expression |
1 |
|
cardid2 |
⊢ ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) ≈ 𝐴 ) |
2 |
|
bren |
⊢ ( ( card ‘ 𝐴 ) ≈ 𝐴 ↔ ∃ 𝑓 𝑓 : ( card ‘ 𝐴 ) –1-1-onto→ 𝐴 ) |
3 |
1 2
|
sylib |
⊢ ( 𝐴 ∈ dom card → ∃ 𝑓 𝑓 : ( card ‘ 𝐴 ) –1-1-onto→ 𝐴 ) |
4 |
|
sqxpexg |
⊢ ( 𝐴 ∈ dom card → ( 𝐴 × 𝐴 ) ∈ V ) |
5 |
|
incom |
⊢ ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ∩ ( 𝐴 × 𝐴 ) ) = ( ( 𝐴 × 𝐴 ) ∩ { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ) |
6 |
|
inex1g |
⊢ ( ( 𝐴 × 𝐴 ) ∈ V → ( ( 𝐴 × 𝐴 ) ∩ { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ) ∈ V ) |
7 |
5 6
|
eqeltrid |
⊢ ( ( 𝐴 × 𝐴 ) ∈ V → ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ∩ ( 𝐴 × 𝐴 ) ) ∈ V ) |
8 |
4 7
|
syl |
⊢ ( 𝐴 ∈ dom card → ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ∩ ( 𝐴 × 𝐴 ) ) ∈ V ) |
9 |
|
f1ocnv |
⊢ ( 𝑓 : ( card ‘ 𝐴 ) –1-1-onto→ 𝐴 → ◡ 𝑓 : 𝐴 –1-1-onto→ ( card ‘ 𝐴 ) ) |
10 |
|
cardon |
⊢ ( card ‘ 𝐴 ) ∈ On |
11 |
10
|
onordi |
⊢ Ord ( card ‘ 𝐴 ) |
12 |
|
ordwe |
⊢ ( Ord ( card ‘ 𝐴 ) → E We ( card ‘ 𝐴 ) ) |
13 |
11 12
|
ax-mp |
⊢ E We ( card ‘ 𝐴 ) |
14 |
|
eqid |
⊢ { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } = { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } |
15 |
14
|
f1owe |
⊢ ( ◡ 𝑓 : 𝐴 –1-1-onto→ ( card ‘ 𝐴 ) → ( E We ( card ‘ 𝐴 ) → { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } We 𝐴 ) ) |
16 |
9 13 15
|
mpisyl |
⊢ ( 𝑓 : ( card ‘ 𝐴 ) –1-1-onto→ 𝐴 → { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } We 𝐴 ) |
17 |
|
weinxp |
⊢ ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } We 𝐴 ↔ ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ∩ ( 𝐴 × 𝐴 ) ) We 𝐴 ) |
18 |
16 17
|
sylib |
⊢ ( 𝑓 : ( card ‘ 𝐴 ) –1-1-onto→ 𝐴 → ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ∩ ( 𝐴 × 𝐴 ) ) We 𝐴 ) |
19 |
|
weeq1 |
⊢ ( 𝑥 = ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ∩ ( 𝐴 × 𝐴 ) ) → ( 𝑥 We 𝐴 ↔ ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ∩ ( 𝐴 × 𝐴 ) ) We 𝐴 ) ) |
20 |
19
|
spcegv |
⊢ ( ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ∩ ( 𝐴 × 𝐴 ) ) ∈ V → ( ( { 〈 𝑧 , 𝑤 〉 ∣ ( ◡ 𝑓 ‘ 𝑧 ) E ( ◡ 𝑓 ‘ 𝑤 ) } ∩ ( 𝐴 × 𝐴 ) ) We 𝐴 → ∃ 𝑥 𝑥 We 𝐴 ) ) |
21 |
8 18 20
|
syl2im |
⊢ ( 𝐴 ∈ dom card → ( 𝑓 : ( card ‘ 𝐴 ) –1-1-onto→ 𝐴 → ∃ 𝑥 𝑥 We 𝐴 ) ) |
22 |
21
|
exlimdv |
⊢ ( 𝐴 ∈ dom card → ( ∃ 𝑓 𝑓 : ( card ‘ 𝐴 ) –1-1-onto→ 𝐴 → ∃ 𝑥 𝑥 We 𝐴 ) ) |
23 |
3 22
|
mpd |
⊢ ( 𝐴 ∈ dom card → ∃ 𝑥 𝑥 We 𝐴 ) |