| Step | Hyp | Ref
| Expression |
| 1 | | cardf2 9983 |
. . . . . . 7
⊢
card:{𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On |
| 2 | | ffun 6739 |
. . . . . . 7
⊢
(card:{𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card) |
| 3 | 1, 2 | ax-mp 5 |
. . . . . 6
⊢ Fun
card |
| 4 | | r1fnon 9807 |
. . . . . . 7
⊢
𝑅1 Fn On |
| 5 | | fnfun 6668 |
. . . . . . 7
⊢
(𝑅1 Fn On → Fun
𝑅1) |
| 6 | 4, 5 | ax-mp 5 |
. . . . . 6
⊢ Fun
𝑅1 |
| 7 | | funco 6606 |
. . . . . 6
⊢ ((Fun
card ∧ Fun 𝑅1) → Fun (card ∘
𝑅1)) |
| 8 | 3, 6, 7 | mp2an 692 |
. . . . 5
⊢ Fun (card
∘ 𝑅1) |
| 9 | | funfn 6596 |
. . . . 5
⊢ (Fun
(card ∘ 𝑅1) ↔ (card ∘
𝑅1) Fn dom (card ∘
𝑅1)) |
| 10 | 8, 9 | mpbi 230 |
. . . 4
⊢ (card
∘ 𝑅1) Fn dom (card ∘
𝑅1) |
| 11 | | rnco 6272 |
. . . . 5
⊢ ran (card
∘ 𝑅1) = ran (card ↾ ran
𝑅1) |
| 12 | | resss 6019 |
. . . . . . 7
⊢ (card
↾ ran 𝑅1) ⊆ card |
| 13 | 12 | rnssi 5951 |
. . . . . 6
⊢ ran (card
↾ ran 𝑅1) ⊆ ran card |
| 14 | | frn 6743 |
. . . . . . 7
⊢
(card:{𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → ran card ⊆
On) |
| 15 | 1, 14 | ax-mp 5 |
. . . . . 6
⊢ ran card
⊆ On |
| 16 | 13, 15 | sstri 3993 |
. . . . 5
⊢ ran (card
↾ ran 𝑅1) ⊆ On |
| 17 | 11, 16 | eqsstri 4030 |
. . . 4
⊢ ran (card
∘ 𝑅1) ⊆ On |
| 18 | | df-f 6565 |
. . . 4
⊢ ((card
∘ 𝑅1):dom (card ∘
𝑅1)⟶On ↔ ((card ∘ 𝑅1)
Fn dom (card ∘ 𝑅1) ∧ ran (card ∘
𝑅1) ⊆ On)) |
| 19 | 10, 17, 18 | mpbir2an 711 |
. . 3
⊢ (card
∘ 𝑅1):dom (card ∘
𝑅1)⟶On |
| 20 | | dmco 6274 |
. . . 4
⊢ dom (card
∘ 𝑅1) = (◡𝑅1 “ dom
card) |
| 21 | 20 | feq2i 6728 |
. . 3
⊢ ((card
∘ 𝑅1):dom (card ∘
𝑅1)⟶On ↔ (card ∘
𝑅1):(◡𝑅1 “ dom
card)⟶On) |
| 22 | 19, 21 | mpbi 230 |
. 2
⊢ (card
∘ 𝑅1):(◡𝑅1 “ dom
card)⟶On |
| 23 | | elpreima 7078 |
. . . . . . . . 9
⊢
(𝑅1 Fn On → (𝑥 ∈ (◡𝑅1 “ dom card)
↔ (𝑥 ∈ On ∧
(𝑅1‘𝑥) ∈ dom card))) |
| 24 | 4, 23 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑥 ∈ (◡𝑅1 “ dom card)
↔ (𝑥 ∈ On ∧
(𝑅1‘𝑥) ∈ dom card)) |
| 25 | 24 | simplbi 497 |
. . . . . . 7
⊢ (𝑥 ∈ (◡𝑅1 “ dom card)
→ 𝑥 ∈
On) |
| 26 | | onelon 6409 |
. . . . . . 7
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) |
| 27 | 25, 26 | sylan 580 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) |
| 28 | 24 | simprbi 496 |
. . . . . . . 8
⊢ (𝑥 ∈ (◡𝑅1 “ dom card)
→ (𝑅1‘𝑥) ∈ dom card) |
| 29 | 28 | adantr 480 |
. . . . . . 7
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(𝑅1‘𝑥) ∈ dom card) |
| 30 | | r1ord2 9821 |
. . . . . . . . 9
⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆
(𝑅1‘𝑥))) |
| 31 | 30 | imp 406 |
. . . . . . . 8
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ⊆
(𝑅1‘𝑥)) |
| 32 | 25, 31 | sylan 580 |
. . . . . . 7
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)) |
| 33 | | ssnum 10079 |
. . . . . . 7
⊢
(((𝑅1‘𝑥) ∈ dom card ∧
(𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)) →
(𝑅1‘𝑦) ∈ dom card) |
| 34 | 29, 32, 33 | syl2anc 584 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(𝑅1‘𝑦) ∈ dom card) |
| 35 | | elpreima 7078 |
. . . . . . 7
⊢
(𝑅1 Fn On → (𝑦 ∈ (◡𝑅1 “ dom card)
↔ (𝑦 ∈ On ∧
(𝑅1‘𝑦) ∈ dom card))) |
| 36 | 4, 35 | ax-mp 5 |
. . . . . 6
⊢ (𝑦 ∈ (◡𝑅1 “ dom card)
↔ (𝑦 ∈ On ∧
(𝑅1‘𝑦) ∈ dom card)) |
| 37 | 27, 34, 36 | sylanbrc 583 |
. . . . 5
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (◡𝑅1 “ dom
card)) |
| 38 | 37 | rgen2 3199 |
. . . 4
⊢
∀𝑥 ∈
(◡𝑅1 “ dom
card)∀𝑦 ∈ 𝑥 𝑦 ∈ (◡𝑅1 “ dom
card) |
| 39 | | dftr5 5263 |
. . . 4
⊢ (Tr
(◡𝑅1 “ dom
card) ↔ ∀𝑥
∈ (◡𝑅1 “
dom card)∀𝑦 ∈
𝑥 𝑦 ∈ (◡𝑅1 “ dom
card)) |
| 40 | 38, 39 | mpbir 231 |
. . 3
⊢ Tr (◡𝑅1 “ dom
card) |
| 41 | | cnvimass 6100 |
. . . . 5
⊢ (◡𝑅1 “ dom card)
⊆ dom 𝑅1 |
| 42 | | dffn2 6738 |
. . . . . . 7
⊢
(𝑅1 Fn On ↔
𝑅1:On⟶V) |
| 43 | 4, 42 | mpbi 230 |
. . . . . 6
⊢
𝑅1:On⟶V |
| 44 | 43 | fdmi 6747 |
. . . . 5
⊢ dom
𝑅1 = On |
| 45 | 41, 44 | sseqtri 4032 |
. . . 4
⊢ (◡𝑅1 “ dom card)
⊆ On |
| 46 | | epweon 7795 |
. . . 4
⊢ E We
On |
| 47 | | wess 5671 |
. . . 4
⊢ ((◡𝑅1 “ dom card)
⊆ On → ( E We On → E We (◡𝑅1 “ dom
card))) |
| 48 | 45, 46, 47 | mp2 9 |
. . 3
⊢ E We
(◡𝑅1 “ dom
card) |
| 49 | | df-ord 6387 |
. . 3
⊢ (Ord
(◡𝑅1 “ dom
card) ↔ (Tr (◡𝑅1 “ dom card)
∧ E We (◡𝑅1
“ dom card))) |
| 50 | 40, 48, 49 | mpbir2an 711 |
. 2
⊢ Ord
(◡𝑅1 “ dom
card) |
| 51 | | r1sdom 9814 |
. . . . . . 7
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺
(𝑅1‘𝑥)) |
| 52 | 25, 51 | sylan 580 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(𝑅1‘𝑦) ≺ (𝑅1‘𝑥)) |
| 53 | | cardsdom2 10028 |
. . . . . . 7
⊢
(((𝑅1‘𝑦) ∈ dom card ∧
(𝑅1‘𝑥) ∈ dom card) →
((card‘(𝑅1‘𝑦)) ∈
(card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺
(𝑅1‘𝑥))) |
| 54 | 34, 29, 53 | syl2anc 584 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
((card‘(𝑅1‘𝑦)) ∈
(card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺
(𝑅1‘𝑥))) |
| 55 | 52, 54 | mpbird 257 |
. . . . 5
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) →
(card‘(𝑅1‘𝑦)) ∈
(card‘(𝑅1‘𝑥))) |
| 56 | | fvco2 7006 |
. . . . . 6
⊢
((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘
𝑅1)‘𝑦) =
(card‘(𝑅1‘𝑦))) |
| 57 | 4, 27, 56 | sylancr 587 |
. . . . 5
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → ((card ∘
𝑅1)‘𝑦) =
(card‘(𝑅1‘𝑦))) |
| 58 | 25 | adantr 480 |
. . . . . 6
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ On) |
| 59 | | fvco2 7006 |
. . . . . 6
⊢
((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘
𝑅1)‘𝑥) =
(card‘(𝑅1‘𝑥))) |
| 60 | 4, 58, 59 | sylancr 587 |
. . . . 5
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → ((card ∘
𝑅1)‘𝑥) =
(card‘(𝑅1‘𝑥))) |
| 61 | 55, 57, 60 | 3eltr4d 2856 |
. . . 4
⊢ ((𝑥 ∈ (◡𝑅1 “ dom card)
∧ 𝑦 ∈ 𝑥) → ((card ∘
𝑅1)‘𝑦) ∈ ((card ∘
𝑅1)‘𝑥)) |
| 62 | 61 | ex 412 |
. . 3
⊢ (𝑥 ∈ (◡𝑅1 “ dom card)
→ (𝑦 ∈ 𝑥 → ((card ∘
𝑅1)‘𝑦) ∈ ((card ∘
𝑅1)‘𝑥))) |
| 63 | 62 | adantl 481 |
. 2
⊢ ((𝑦 ∈ (◡𝑅1 “ dom card)
∧ 𝑥 ∈ (◡𝑅1 “ dom card))
→ (𝑦 ∈ 𝑥 → ((card ∘
𝑅1)‘𝑦) ∈ ((card ∘
𝑅1)‘𝑥))) |
| 64 | 22, 50, 63, 20 | issmo 8388 |
1
⊢ Smo (card
∘ 𝑅1) |