Step | Hyp | Ref
| Expression |
1 | | cardf2 8652 |
. . . . . . 7
⊢
card:{𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On |
2 | | ffun 5961 |
. . . . . . 7
⊢
(card:{𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card) |
3 | 1, 2 | ax-mp 5 |
. . . . . 6
⊢ Fun
card |
4 | | r1fnon 8513 |
. . . . . . 7
⊢
𝑅1 Fn On |
5 | | fnfun 5902 |
. . . . . . 7
⊢
(𝑅1 Fn On → Fun
𝑅1) |
6 | 4, 5 | ax-mp 5 |
. . . . . 6
⊢ Fun
𝑅1 |
7 | | funco 5842 |
. . . . . 6
⊢ ((Fun
card ∧ Fun 𝑅1) → Fun (card ∘
𝑅1)) |
8 | 3, 6, 7 | mp2an 704 |
. . . . 5
⊢ Fun (card
∘ 𝑅1) |
9 | | funfn 5833 |
. . . . 5
⊢ (Fun
(card ∘ 𝑅1) ↔ (card ∘
𝑅1) Fn dom (card ∘
𝑅1)) |
10 | 8, 9 | mpbi 219 |
. . . 4
⊢ (card
∘ 𝑅1) Fn dom (card ∘
𝑅1) |
11 | | rnco 5558 |
. . . . 5
⊢ ran (card
∘ 𝑅1) = ran (card ↾ ran
𝑅1) |
12 | | resss 5342 |
. . . . . . 7
⊢ (card
↾ ran 𝑅1) ⊆ card |
13 | | rnss 5275 |
. . . . . . 7
⊢ ((card
↾ ran 𝑅1) ⊆ card → ran (card ↾ ran
𝑅1) ⊆ ran card) |
14 | 12, 13 | ax-mp 5 |
. . . . . 6
⊢ ran (card
↾ ran 𝑅1) ⊆ ran card |
15 | | frn 5966 |
. . . . . . 7
⊢
(card:{𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → ran card ⊆
On) |
16 | 1, 15 | ax-mp 5 |
. . . . . 6
⊢ ran card
⊆ On |
17 | 14, 16 | sstri 3577 |
. . . . 5
⊢ ran (card
↾ ran 𝑅1) ⊆ On |
18 | 11, 17 | eqsstri 3598 |
. . . 4
⊢ ran (card
∘ 𝑅1) ⊆ On |
19 | | df-f 5808 |
. . . 4
⊢ ((card
∘ 𝑅1):dom (card ∘
𝑅1)⟶On ↔ ((card ∘ 𝑅1)
Fn dom (card ∘ 𝑅1) ∧ ran (card ∘
𝑅1) ⊆ On)) |
20 | 10, 18, 19 | mpbir2an 957 |
. . 3
⊢ (card
∘ 𝑅1):dom (card ∘
𝑅1)⟶On |
21 | | dmco 5560 |
. . . 4
⊢ dom (card
∘ 𝑅1) = (◡𝑅1 “ dom
card) |
22 | 21 | feq2i 5950 |
. . 3
⊢ ((card
∘ 𝑅1):dom (card ∘
𝑅1)⟶On ↔ (card ∘
𝑅1):(◡𝑅1 “ dom
card)⟶On) |
23 | 20, 22 | mpbi 219 |
. 2
⊢ (card
∘ 𝑅1):(◡𝑅1 “ dom
card)⟶On |
24 | | elpreima 6245 |
. . . . . . . . 9
⊢
(𝑅1 Fn On → (𝑥 ∈ (◡𝑅1 “ dom card)
↔ (𝑥 ∈ On ∧
(𝑅1‘𝑥) ∈ dom card))) |
25 | 4, 24 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑥 ∈ (◡𝑅1 “ dom card)
↔ (𝑥 ∈ On ∧
(𝑅1‘𝑥) ∈ dom card)) |
26 | 25 | simplbi 475 |
. . . . . . 7
⊢ (𝑥 ∈ (◡𝑅1 “ dom card)
→ 𝑥 ∈
On) |
27 | | onelon 5665 |
. . . . . . 7
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) |
28 | 26, 27 | sylan 487 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) |
29 | 25 | simprbi 479 |
. . . . . . . 8
⊢ (𝑥 ∈ (◡𝑅1 “ dom card)
→ (𝑅1‘𝑥) ∈ dom card) |
30 | 29 | adantr 480 |
. . . . . . 7
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(𝑅1‘𝑥) ∈ dom card) |
31 | | r1ord2 8527 |
. . . . . . . . 9
⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆
(𝑅1‘𝑥))) |
32 | 31 | imp 444 |
. . . . . . . 8
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ⊆
(𝑅1‘𝑥)) |
33 | 26, 32 | sylan 487 |
. . . . . . 7
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)) |
34 | | ssnum 8745 |
. . . . . . 7
⊢
(((𝑅1‘𝑥) ∈ dom card ∧
(𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)) →
(𝑅1‘𝑦) ∈ dom card) |
35 | 30, 33, 34 | syl2anc 691 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(𝑅1‘𝑦) ∈ dom card) |
36 | | elpreima 6245 |
. . . . . . 7
⊢
(𝑅1 Fn On → (𝑦 ∈ (◡𝑅1 “ dom card)
↔ (𝑦 ∈ On ∧
(𝑅1‘𝑦) ∈ dom card))) |
37 | 4, 36 | ax-mp 5 |
. . . . . 6
⊢ (𝑦 ∈ (◡𝑅1 “ dom card)
↔ (𝑦 ∈ On ∧
(𝑅1‘𝑦) ∈ dom card)) |
38 | 28, 35, 37 | sylanbrc 695 |
. . . . 5
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (◡𝑅1 “ dom
card)) |
39 | 38 | rgen2 2958 |
. . . 4
⊢
∀𝑥 ∈
(◡𝑅1 “ dom
card)∀𝑦 ∈ 𝑥 𝑦 ∈ (◡𝑅1 “ dom
card) |
40 | | dftr5 4683 |
. . . 4
⊢ (Tr
(◡𝑅1 “ dom
card) ↔ ∀𝑥
∈ (◡𝑅1 “
dom card)∀𝑦 ∈
𝑥 𝑦 ∈ (◡𝑅1 “ dom
card)) |
41 | 39, 40 | mpbir 220 |
. . 3
⊢ Tr (◡𝑅1 “ dom
card) |
42 | | cnvimass 5404 |
. . . . 5
⊢ (◡𝑅1 “ dom card)
⊆ dom 𝑅1 |
43 | | dffn2 5960 |
. . . . . . 7
⊢
(𝑅1 Fn On ↔
𝑅1:On⟶V) |
44 | 4, 43 | mpbi 219 |
. . . . . 6
⊢
𝑅1:On⟶V |
45 | 44 | fdmi 5965 |
. . . . 5
⊢ dom
𝑅1 = On |
46 | 42, 45 | sseqtri 3600 |
. . . 4
⊢ (◡𝑅1 “ dom card)
⊆ On |
47 | | epweon 6875 |
. . . 4
⊢ E We
On |
48 | | wess 5025 |
. . . 4
⊢ ((◡𝑅1 “ dom card)
⊆ On → ( E We On → E We (◡𝑅1 “ dom
card))) |
49 | 46, 47, 48 | mp2 9 |
. . 3
⊢ E We
(◡𝑅1 “ dom
card) |
50 | | df-ord 5643 |
. . 3
⊢ (Ord
(◡𝑅1 “ dom
card) ↔ (Tr (◡𝑅1 “ dom card)
∧ E We (◡𝑅1
“ dom card))) |
51 | 41, 49, 50 | mpbir2an 957 |
. 2
⊢ Ord
(◡𝑅1 “ dom
card) |
52 | | r1sdom 8520 |
. . . . . . 7
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺
(𝑅1‘𝑥)) |
53 | 26, 52 | sylan 487 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(𝑅1‘𝑦) ≺ (𝑅1‘𝑥)) |
54 | | cardsdom2 8697 |
. . . . . . 7
⊢
(((𝑅1‘𝑦) ∈ dom card ∧
(𝑅1‘𝑥) ∈ dom card) →
((card‘(𝑅1‘𝑦)) ∈
(card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺
(𝑅1‘𝑥))) |
55 | 35, 30, 54 | syl2anc 691 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
((card‘(𝑅1‘𝑦)) ∈
(card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺
(𝑅1‘𝑥))) |
56 | 53, 55 | mpbird 246 |
. . . . 5
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(card‘(𝑅1‘𝑦)) ∈
(card‘(𝑅1‘𝑥))) |
57 | | fvco2 6183 |
. . . . . 6
⊢
((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘
𝑅1)‘𝑦) =
(card‘(𝑅1‘𝑦))) |
58 | 4, 28, 57 | sylancr 694 |
. . . . 5
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → ((card ∘
𝑅1)‘𝑦) =
(card‘(𝑅1‘𝑦))) |
59 | 26 | adantr 480 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ On) |
60 | | fvco2 6183 |
. . . . . 6
⊢
((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘
𝑅1)‘𝑥) =
(card‘(𝑅1‘𝑥))) |
61 | 4, 59, 60 | sylancr 694 |
. . . . 5
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → ((card ∘
𝑅1)‘𝑥) =
(card‘(𝑅1‘𝑥))) |
62 | 56, 58, 61 | 3eltr4d 2703 |
. . . 4
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → ((card ∘
𝑅1)‘𝑦) ∈ ((card ∘
𝑅1)‘𝑥)) |
63 | 62 | ex 449 |
. . 3
⊢ (𝑥 ∈ (◡𝑅1 “ dom card)
→ (𝑦 ∈ 𝑥 → ((card ∘
𝑅1)‘𝑦) ∈ ((card ∘
𝑅1)‘𝑥))) |
64 | 63 | adantl 481 |
. 2
⊢ ((𝑦 ∈ (◡𝑅1 “ dom card)
∧ 𝑥 ∈ (◡𝑅1 “ dom card))
→ (𝑦 ∈ 𝑥 → ((card ∘
𝑅1)‘𝑦) ∈ ((card ∘
𝑅1)‘𝑥))) |
65 | 23, 51, 64, 21 | issmo 7332 |
1
⊢ Smo (card
∘ 𝑅1) |