| Step | Hyp | Ref
| Expression |
| 1 | | bren 8995 |
. 2
⊢ (suc
𝐴 ≈ suc 𝐵 ↔ ∃𝑓 𝑓:suc 𝐴–1-1-onto→suc
𝐵) |
| 2 | | f1of1 6847 |
. . . . . . . . . 10
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → 𝑓:suc 𝐴–1-1→suc 𝐵) |
| 3 | 2 | adantl 481 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵) → 𝑓:suc 𝐴–1-1→suc 𝐵) |
| 4 | | phplem2OLD.2 |
. . . . . . . . . 10
⊢ 𝐵 ∈ V |
| 5 | 4 | sucex 7826 |
. . . . . . . . 9
⊢ suc 𝐵 ∈ V |
| 6 | | sssucid 6464 |
. . . . . . . . . 10
⊢ 𝐴 ⊆ suc 𝐴 |
| 7 | | phplem2OLD.1 |
. . . . . . . . . 10
⊢ 𝐴 ∈ V |
| 8 | | f1imaen2g 9055 |
. . . . . . . . . 10
⊢ (((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ suc 𝐵 ∈ V) ∧ (𝐴 ⊆ suc 𝐴 ∧ 𝐴 ∈ V)) → (𝑓 “ 𝐴) ≈ 𝐴) |
| 9 | 6, 7, 8 | mpanr12 705 |
. . . . . . . . 9
⊢ ((𝑓:suc 𝐴–1-1→suc 𝐵 ∧ suc 𝐵 ∈ V) → (𝑓 “ 𝐴) ≈ 𝐴) |
| 10 | 3, 5, 9 | sylancl 586 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵) → (𝑓 “ 𝐴) ≈ 𝐴) |
| 11 | 10 | ensymd 9045 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵) → 𝐴 ≈ (𝑓 “ 𝐴)) |
| 12 | | nnord 7895 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ω → Ord 𝐴) |
| 13 | | orddif 6480 |
. . . . . . . . . 10
⊢ (Ord
𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
| 14 | 12, 13 | syl 17 |
. . . . . . . . 9
⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
| 15 | 14 | imaeq2d 6078 |
. . . . . . . 8
⊢ (𝐴 ∈ ω → (𝑓 “ 𝐴) = (𝑓 “ (suc 𝐴 ∖ {𝐴}))) |
| 16 | | f1ofn 6849 |
. . . . . . . . . . 11
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → 𝑓 Fn suc 𝐴) |
| 17 | 7 | sucid 6466 |
. . . . . . . . . . 11
⊢ 𝐴 ∈ suc 𝐴 |
| 18 | | fnsnfv 6988 |
. . . . . . . . . . 11
⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → {(𝑓‘𝐴)} = (𝑓 “ {𝐴})) |
| 19 | 16, 17, 18 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → {(𝑓‘𝐴)} = (𝑓 “ {𝐴})) |
| 20 | 19 | difeq2d 4126 |
. . . . . . . . 9
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴}))) |
| 21 | | imadmrn 6088 |
. . . . . . . . . . . 12
⊢ (𝑓 “ dom 𝑓) = ran 𝑓 |
| 22 | 21 | eqcomi 2746 |
. . . . . . . . . . 11
⊢ ran 𝑓 = (𝑓 “ dom 𝑓) |
| 23 | | f1ofo 6855 |
. . . . . . . . . . . 12
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → 𝑓:suc 𝐴–onto→suc 𝐵) |
| 24 | | forn 6823 |
. . . . . . . . . . . 12
⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ran 𝑓 = suc 𝐵) |
| 25 | 23, 24 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → ran 𝑓 = suc 𝐵) |
| 26 | | f1odm 6852 |
. . . . . . . . . . . 12
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → dom 𝑓 = suc 𝐴) |
| 27 | 26 | imaeq2d 6078 |
. . . . . . . . . . 11
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → (𝑓 “ dom 𝑓) = (𝑓 “ suc 𝐴)) |
| 28 | 22, 25, 27 | 3eqtr3a 2801 |
. . . . . . . . . 10
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → suc 𝐵 = (𝑓 “ suc 𝐴)) |
| 29 | 28 | difeq1d 4125 |
. . . . . . . . 9
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → (suc 𝐵 ∖ {(𝑓‘𝐴)}) = ((𝑓 “ suc 𝐴) ∖ {(𝑓‘𝐴)})) |
| 30 | | dff1o3 6854 |
. . . . . . . . . 10
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 ↔ (𝑓:suc 𝐴–onto→suc 𝐵 ∧ Fun ◡𝑓)) |
| 31 | | imadif 6650 |
. . . . . . . . . 10
⊢ (Fun
◡𝑓 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴}))) |
| 32 | 30, 31 | simplbiim 504 |
. . . . . . . . 9
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = ((𝑓 “ suc 𝐴) ∖ (𝑓 “ {𝐴}))) |
| 33 | 20, 29, 32 | 3eqtr4rd 2788 |
. . . . . . . 8
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → (𝑓 “ (suc 𝐴 ∖ {𝐴})) = (suc 𝐵 ∖ {(𝑓‘𝐴)})) |
| 34 | 15, 33 | sylan9eq 2797 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵) → (𝑓 “ 𝐴) = (suc 𝐵 ∖ {(𝑓‘𝐴)})) |
| 35 | 11, 34 | breqtrd 5169 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵) → 𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)})) |
| 36 | | fnfvelrn 7100 |
. . . . . . . . . 10
⊢ ((𝑓 Fn suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → (𝑓‘𝐴) ∈ ran 𝑓) |
| 37 | 16, 17, 36 | sylancl 586 |
. . . . . . . . 9
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → (𝑓‘𝐴) ∈ ran 𝑓) |
| 38 | 24 | eleq2d 2827 |
. . . . . . . . . 10
⊢ (𝑓:suc 𝐴–onto→suc 𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵)) |
| 39 | 23, 38 | syl 17 |
. . . . . . . . 9
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → ((𝑓‘𝐴) ∈ ran 𝑓 ↔ (𝑓‘𝐴) ∈ suc 𝐵)) |
| 40 | 37, 39 | mpbid 232 |
. . . . . . . 8
⊢ (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → (𝑓‘𝐴) ∈ suc 𝐵) |
| 41 | | fvex 6919 |
. . . . . . . . 9
⊢ (𝑓‘𝐴) ∈ V |
| 42 | 4, 41 | phplem3OLD 9256 |
. . . . . . . 8
⊢ ((𝐵 ∈ ω ∧ (𝑓‘𝐴) ∈ suc 𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)})) |
| 43 | 40, 42 | sylan2 593 |
. . . . . . 7
⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵) → 𝐵 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)})) |
| 44 | 43 | ensymd 9045 |
. . . . . 6
⊢ ((𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵) → (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) |
| 45 | | entr 9046 |
. . . . . 6
⊢ ((𝐴 ≈ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ∧ (suc 𝐵 ∖ {(𝑓‘𝐴)}) ≈ 𝐵) → 𝐴 ≈ 𝐵) |
| 46 | 35, 44, 45 | syl2an 596 |
. . . . 5
⊢ (((𝐴 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵) ∧ (𝐵 ∈ ω ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵)) → 𝐴 ≈ 𝐵) |
| 47 | 46 | anandirs 679 |
. . . 4
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑓:suc 𝐴–1-1-onto→suc
𝐵) → 𝐴 ≈ 𝐵) |
| 48 | 47 | ex 412 |
. . 3
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑓:suc 𝐴–1-1-onto→suc
𝐵 → 𝐴 ≈ 𝐵)) |
| 49 | 48 | exlimdv 1933 |
. 2
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) →
(∃𝑓 𝑓:suc 𝐴–1-1-onto→suc
𝐵 → 𝐴 ≈ 𝐵)) |
| 50 | 1, 49 | biimtrid 242 |
1
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc
𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵)) |