Step | Hyp | Ref
| Expression |
1 | | ssid 3624 |
. 2
⊢ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} |
2 | | hashf1lem2.2 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ Fin) |
3 | | hashf1lem2.1 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ Fin) |
4 | | mapfi 8262 |
. . . . 5
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝐵 ↑𝑚
𝐴) ∈
Fin) |
5 | 2, 3, 4 | syl2anc 693 |
. . . 4
⊢ (𝜑 → (𝐵 ↑𝑚 𝐴) ∈ Fin) |
6 | | f1f 6101 |
. . . . . 6
⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) |
7 | 2, 3 | elmapd 7871 |
. . . . . 6
⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝐵)) |
8 | 6, 7 | syl5ibr 236 |
. . . . 5
⊢ (𝜑 → (𝑓:𝐴–1-1→𝐵 → 𝑓 ∈ (𝐵 ↑𝑚 𝐴))) |
9 | 8 | abssdv 3676 |
. . . 4
⊢ (𝜑 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ (𝐵 ↑𝑚 𝐴)) |
10 | | ssfi 8180 |
. . . 4
⊢ (((𝐵 ↑𝑚
𝐴) ∈ Fin ∧ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ (𝐵 ↑𝑚 𝐴)) → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ Fin) |
11 | 5, 9, 10 | syl2anc 693 |
. . 3
⊢ (𝜑 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ Fin) |
12 | | sseq1 3626 |
. . . . . 6
⊢ (𝑥 = ∅ → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ ∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) |
13 | | eleq2 2690 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ∅ → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ ∅)) |
14 | | noel 3919 |
. . . . . . . . . . . . . 14
⊢ ¬
(𝑓 ↾ 𝐴) ∈
∅ |
15 | 14 | pm2.21i 116 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ↾ 𝐴) ∈ ∅ → 𝑓 ∈ ∅) |
16 | 13, 15 | syl6bi 243 |
. . . . . . . . . . . 12
⊢ (𝑥 = ∅ → ((𝑓 ↾ 𝐴) ∈ 𝑥 → 𝑓 ∈ ∅)) |
17 | 16 | adantrd 484 |
. . . . . . . . . . 11
⊢ (𝑥 = ∅ → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓 ∈ ∅)) |
18 | 17 | abssdv 3676 |
. . . . . . . . . 10
⊢ (𝑥 = ∅ → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ ∅) |
19 | | ss0 3974 |
. . . . . . . . . 10
⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ ∅ → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = ∅) |
20 | 18, 19 | syl 17 |
. . . . . . . . 9
⊢ (𝑥 = ∅ → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = ∅) |
21 | 20 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑥 = ∅ →
(#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (#‘∅)) |
22 | | hash0 13158 |
. . . . . . . 8
⊢
(#‘∅) = 0 |
23 | 21, 22 | syl6eq 2672 |
. . . . . . 7
⊢ (𝑥 = ∅ →
(#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = 0) |
24 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑥 = ∅ → (#‘𝑥) =
(#‘∅)) |
25 | 24, 22 | syl6eq 2672 |
. . . . . . . 8
⊢ (𝑥 = ∅ → (#‘𝑥) = 0) |
26 | 25 | oveq2d 6666 |
. . . . . . 7
⊢ (𝑥 = ∅ →
(((#‘𝐵) −
(#‘𝐴)) ·
(#‘𝑥)) =
(((#‘𝐵) −
(#‘𝐴)) ·
0)) |
27 | 23, 26 | eqeq12d 2637 |
. . . . . 6
⊢ (𝑥 = ∅ →
((#‘{𝑓 ∣
((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)) ↔ 0 = (((#‘𝐵) − (#‘𝐴)) · 0))) |
28 | 12, 27 | imbi12d 334 |
. . . . 5
⊢ (𝑥 = ∅ → ((𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥))) ↔ (∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 0 = (((#‘𝐵) − (#‘𝐴)) · 0)))) |
29 | 28 | imbi2d 330 |
. . . 4
⊢ (𝑥 = ∅ → ((𝜑 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)))) ↔ (𝜑 → (∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 0 = (((#‘𝐵) − (#‘𝐴)) · 0))))) |
30 | | sseq1 3626 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) |
31 | | eleq2 2690 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ 𝑦)) |
32 | 31 | anbi1d 741 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))) |
33 | 32 | abbidv 2741 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) |
34 | 33 | fveq2d 6195 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) |
35 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦)) |
36 | 35 | oveq2d 6666 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦))) |
37 | 34, 36 | eqeq12d 2637 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)) ↔ (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)))) |
38 | 30, 37 | imbi12d 334 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥))) ↔ (𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦))))) |
39 | 38 | imbi2d 330 |
. . . 4
⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)))) ↔ (𝜑 → (𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)))))) |
40 | | sseq1 3626 |
. . . . . 6
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) |
41 | | eleq2 2690 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}))) |
42 | 41 | anbi1d 741 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))) |
43 | 42 | abbidv 2741 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) |
44 | 43 | fveq2d 6195 |
. . . . . . 7
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) |
45 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (#‘𝑥) = (#‘(𝑦 ∪ {𝑎}))) |
46 | 45 | oveq2d 6666 |
. . . . . . 7
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎})))) |
47 | 44, 46 | eqeq12d 2637 |
. . . . . 6
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)) ↔ (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎}))))) |
48 | 40, 47 | imbi12d 334 |
. . . . 5
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥))) ↔ ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎})))))) |
49 | 48 | imbi2d 330 |
. . . 4
⊢ (𝑥 = (𝑦 ∪ {𝑎}) → ((𝜑 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)))) ↔ (𝜑 → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎}))))))) |
50 | | sseq1 3626 |
. . . . . 6
⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) |
51 | | f1eq1 6096 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑦 → (𝑓:𝐴–1-1→𝐵 ↔ 𝑦:𝐴–1-1→𝐵)) |
52 | 51 | cbvabv 2747 |
. . . . . . . . . 10
⊢ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} |
53 | 52 | eqeq2i 2634 |
. . . . . . . . 9
⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵}) |
54 | | ssun1 3776 |
. . . . . . . . . . . . . . 15
⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧}) |
55 | | f1ssres 6108 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → (𝑓 ↾ 𝐴):𝐴–1-1→𝐵) |
56 | 54, 55 | mpan2 707 |
. . . . . . . . . . . . . 14
⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴):𝐴–1-1→𝐵) |
57 | | vex 3203 |
. . . . . . . . . . . . . . . 16
⊢ 𝑓 ∈ V |
58 | 57 | resex 5443 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ↾ 𝐴) ∈ V |
59 | | f1eq1 6096 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑓 ↾ 𝐴) → (𝑦:𝐴–1-1→𝐵 ↔ (𝑓 ↾ 𝐴):𝐴–1-1→𝐵)) |
60 | 58, 59 | elab 3350 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} ↔ (𝑓 ↾ 𝐴):𝐴–1-1→𝐵) |
61 | 56, 60 | sylibr 224 |
. . . . . . . . . . . . 13
⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵}) |
62 | | eleq2 2690 |
. . . . . . . . . . . . 13
⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → ((𝑓 ↾ 𝐴) ∈ 𝑥 ↔ (𝑓 ↾ 𝐴) ∈ {𝑦 ∣ 𝑦:𝐴–1-1→𝐵})) |
63 | 61, 62 | syl5ibr 236 |
. . . . . . . . . . . 12
⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → (𝑓 ↾ 𝐴) ∈ 𝑥)) |
64 | 63 | pm4.71rd 667 |
. . . . . . . . . . 11
⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))) |
65 | 64 | bicomd 213 |
. . . . . . . . . 10
⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → (((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) |
66 | 65 | abbidv 2741 |
. . . . . . . . 9
⊢ (𝑥 = {𝑦 ∣ 𝑦:𝐴–1-1→𝐵} → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) |
67 | 53, 66 | sylbi 207 |
. . . . . . . 8
⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) |
68 | 67 | fveq2d 6195 |
. . . . . . 7
⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (#‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵})) |
69 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘𝑥) = (#‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) |
70 | 69 | oveq2d 6666 |
. . . . . . 7
⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))) |
71 | 68, 70 | eqeq12d 2637 |
. . . . . 6
⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)) ↔ (#‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))) |
72 | 50, 71 | imbi12d 334 |
. . . . 5
⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → ((𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥))) ↔ ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))))) |
73 | 72 | imbi2d 330 |
. . . 4
⊢ (𝑥 = {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → ((𝜑 → (𝑥 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑥 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑥)))) ↔ (𝜑 → ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))))) |
74 | | hashcl 13147 |
. . . . . . . . . 10
⊢ (𝐵 ∈ Fin →
(#‘𝐵) ∈
ℕ0) |
75 | 2, 74 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (#‘𝐵) ∈
ℕ0) |
76 | 75 | nn0cnd 11353 |
. . . . . . . 8
⊢ (𝜑 → (#‘𝐵) ∈ ℂ) |
77 | | hashcl 13147 |
. . . . . . . . . 10
⊢ (𝐴 ∈ Fin →
(#‘𝐴) ∈
ℕ0) |
78 | 3, 77 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (#‘𝐴) ∈
ℕ0) |
79 | 78 | nn0cnd 11353 |
. . . . . . . 8
⊢ (𝜑 → (#‘𝐴) ∈ ℂ) |
80 | 76, 79 | subcld 10392 |
. . . . . . 7
⊢ (𝜑 → ((#‘𝐵) − (#‘𝐴)) ∈
ℂ) |
81 | 80 | mul01d 10235 |
. . . . . 6
⊢ (𝜑 → (((#‘𝐵) − (#‘𝐴)) · 0) =
0) |
82 | 81 | eqcomd 2628 |
. . . . 5
⊢ (𝜑 → 0 = (((#‘𝐵) − (#‘𝐴)) · 0)) |
83 | 82 | a1d 25 |
. . . 4
⊢ (𝜑 → (∅ ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 0 = (((#‘𝐵) − (#‘𝐴)) · 0))) |
84 | | ssun1 3776 |
. . . . . . . . 9
⊢ 𝑦 ⊆ (𝑦 ∪ {𝑎}) |
85 | | sstr 3611 |
. . . . . . . . 9
⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑎}) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) |
86 | 84, 85 | mpan 706 |
. . . . . . . 8
⊢ ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → 𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) |
87 | 86 | imim1i 63 |
. . . . . . 7
⊢ ((𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦))) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)))) |
88 | | oveq1 6657 |
. . . . . . . . . 10
⊢
((#‘{𝑓 ∣
((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)) → ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + ((#‘𝐵) − (#‘𝐴))) = ((((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)) + ((#‘𝐵) − (#‘𝐴)))) |
89 | | elun 3753 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) ∈ {𝑎})) |
90 | 58 | elsn 4192 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓 ↾ 𝐴) ∈ {𝑎} ↔ (𝑓 ↾ 𝐴) = 𝑎) |
91 | 90 | orbi2i 541 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) ∈ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎)) |
92 | 89, 91 | bitri 264 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ↔ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎)) |
93 | 92 | anbi1i 731 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) |
94 | | andir 912 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∨ (𝑓 ↾ 𝐴) = 𝑎) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∨ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))) |
95 | 93, 94 | bitri 264 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∨ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))) |
96 | 95 | abbii 2739 |
. . . . . . . . . . . . . . 15
⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∨ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} |
97 | | unab 3894 |
. . . . . . . . . . . . . . 15
⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∨ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} |
98 | 96, 97 | eqtr4i 2647 |
. . . . . . . . . . . . . 14
⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} = ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) |
99 | 98 | fveq2i 6194 |
. . . . . . . . . . . . 13
⊢
(#‘{𝑓 ∣
((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (#‘({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) |
100 | | snfi 8038 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑧} ∈ Fin |
101 | | unfi 8227 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin) |
102 | 3, 100, 101 | sylancl 694 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin) |
103 | | mapvalg 7867 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∪ {𝑧}) ∈ Fin) → (𝐵 ↑𝑚 (𝐴 ∪ {𝑧})) = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}) |
104 | 2, 102, 103 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐵 ↑𝑚 (𝐴 ∪ {𝑧})) = {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}) |
105 | | mapfi 8262 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∪ {𝑧}) ∈ Fin) → (𝐵 ↑𝑚 (𝐴 ∪ {𝑧})) ∈ Fin) |
106 | 2, 102, 105 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐵 ↑𝑚 (𝐴 ∪ {𝑧})) ∈ Fin) |
107 | 104, 106 | eqeltrrd 2702 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} ∈ Fin) |
108 | | f1f 6101 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵) |
109 | 108 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵) |
110 | 109 | ss2abi 3674 |
. . . . . . . . . . . . . . . 16
⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} |
111 | | ssfi 8180 |
. . . . . . . . . . . . . . . 16
⊢ (({𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} ∈ Fin ∧ {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}) → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin) |
112 | 107, 110,
111 | sylancl 694 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin) |
113 | 112 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → {𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin) |
114 | 108 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵) |
115 | 114 | ss2abi 3674 |
. . . . . . . . . . . . . . . 16
⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} |
116 | | ssfi 8180 |
. . . . . . . . . . . . . . . 16
⊢ (({𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵} ∈ Fin ∧ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵}) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin) |
117 | 107, 115,
116 | sylancl 694 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin) |
118 | 117 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin) |
119 | | inab 3895 |
. . . . . . . . . . . . . . 15
⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∩ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} |
120 | | simprlr 803 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ¬ 𝑎 ∈ 𝑦) |
121 | | abn0 3954 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} ≠ ∅ ↔ ∃𝑓(((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))) |
122 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑓 ↾ 𝐴) = 𝑎) |
123 | | simpll 790 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑓 ↾ 𝐴) ∈ 𝑦) |
124 | 122, 123 | eqeltrrd 2702 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑎 ∈ 𝑦) |
125 | 124 | exlimiv 1858 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑓(((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑎 ∈ 𝑦) |
126 | 121, 125 | sylbi 207 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} ≠ ∅ → 𝑎 ∈ 𝑦) |
127 | 126 | necon1bi 2822 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑎 ∈ 𝑦 → {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} = ∅) |
128 | 120, 127 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → {𝑓 ∣ (((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵))} = ∅) |
129 | 119, 128 | syl5eq 2668 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∩ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ∅) |
130 | | hashun 13171 |
. . . . . . . . . . . . . 14
⊢ (({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin ∧ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin ∧ ({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∩ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ∅) → (#‘({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))) |
131 | 113, 118,
129, 130 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (#‘({𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∪ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))) |
132 | 99, 131 | syl5eq 2668 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))) |
133 | | simpr 477 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) |
134 | 133 | unssbd 3791 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → {𝑎} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) |
135 | | vex 3203 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑎 ∈ V |
136 | 135 | snss 4316 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ {𝑎} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) |
137 | 134, 136 | sylibr 224 |
. . . . . . . . . . . . . . 15
⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) |
138 | | f1eq1 6096 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = 𝑎 → (𝑓:𝐴–1-1→𝐵 ↔ 𝑎:𝐴–1-1→𝐵)) |
139 | 135, 138 | elab 3350 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ↔ 𝑎:𝐴–1-1→𝐵) |
140 | 137, 139 | sylib 208 |
. . . . . . . . . . . . . 14
⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) → 𝑎:𝐴–1-1→𝐵) |
141 | 79 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (#‘𝐴) ∈ ℂ) |
142 | 117 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin) |
143 | | hashcl 13147 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ Fin → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) ∈
ℕ0) |
144 | 142, 143 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) ∈
ℕ0) |
145 | 144 | nn0cnd 11353 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) ∈ ℂ) |
146 | 141, 145 | pncan2d 10394 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (((#‘𝐴) + (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) − (#‘𝐴)) = (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) |
147 | | f1f1orn 6148 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎:𝐴–1-1→𝐵 → 𝑎:𝐴–1-1-onto→ran
𝑎) |
148 | 147 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝑎:𝐴–1-1-onto→ran
𝑎) |
149 | | f1oen3g 7971 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑎 ∈ V ∧ 𝑎:𝐴–1-1-onto→ran
𝑎) → 𝐴 ≈ ran 𝑎) |
150 | 135, 148,
149 | sylancr 695 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐴 ≈ ran 𝑎) |
151 | | hasheni 13136 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ≈ ran 𝑎 → (#‘𝐴) = (#‘ran 𝑎)) |
152 | 150, 151 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (#‘𝐴) = (#‘ran 𝑎)) |
153 | 3 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐴 ∈ Fin) |
154 | 2 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝐵 ∈ Fin) |
155 | | hashf1lem2.3 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴) |
156 | 155 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ¬ 𝑧 ∈ 𝐴) |
157 | | hashf1lem2.4 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵)) |
158 | 157 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((#‘𝐴) + 1) ≤ (#‘𝐵)) |
159 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → 𝑎:𝐴–1-1→𝐵) |
160 | 153, 154,
156, 158, 159 | hashf1lem1 13239 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝑎)) |
161 | | hasheni 13136 |
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝑎) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (#‘(𝐵 ∖ ran 𝑎))) |
162 | 160, 161 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (#‘(𝐵 ∖ ran 𝑎))) |
163 | 152, 162 | oveq12d 6668 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((#‘𝐴) + (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = ((#‘ran 𝑎) + (#‘(𝐵 ∖ ran 𝑎)))) |
164 | | f1f 6101 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎:𝐴–1-1→𝐵 → 𝑎:𝐴⟶𝐵) |
165 | | frn 6053 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎:𝐴⟶𝐵 → ran 𝑎 ⊆ 𝐵) |
166 | 164, 165 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎:𝐴–1-1→𝐵 → ran 𝑎 ⊆ 𝐵) |
167 | 166 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ran 𝑎 ⊆ 𝐵) |
168 | | ssfi 8180 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐵 ∈ Fin ∧ ran 𝑎 ⊆ 𝐵) → ran 𝑎 ∈ Fin) |
169 | 154, 167,
168 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ran 𝑎 ∈ Fin) |
170 | | diffi 8192 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐵 ∈ Fin → (𝐵 ∖ ran 𝑎) ∈ Fin) |
171 | 154, 170 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (𝐵 ∖ ran 𝑎) ∈ Fin) |
172 | | disjdif 4040 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ran
𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅ |
173 | 172 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (ran 𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅) |
174 | | hashun 13171 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ran
𝑎 ∈ Fin ∧ (𝐵 ∖ ran 𝑎) ∈ Fin ∧ (ran 𝑎 ∩ (𝐵 ∖ ran 𝑎)) = ∅) → (#‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = ((#‘ran 𝑎) + (#‘(𝐵 ∖ ran 𝑎)))) |
175 | 169, 171,
173, 174 | syl3anc 1326 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (#‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = ((#‘ran 𝑎) + (#‘(𝐵 ∖ ran 𝑎)))) |
176 | | undif 4049 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ran
𝑎 ⊆ 𝐵 ↔ (ran 𝑎 ∪ (𝐵 ∖ ran 𝑎)) = 𝐵) |
177 | 167, 176 | sylib 208 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (ran 𝑎 ∪ (𝐵 ∖ ran 𝑎)) = 𝐵) |
178 | 177 | fveq2d 6195 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (#‘(ran 𝑎 ∪ (𝐵 ∖ ran 𝑎))) = (#‘𝐵)) |
179 | 163, 175,
178 | 3eqtr2d 2662 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → ((#‘𝐴) + (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = (#‘𝐵)) |
180 | 179 | oveq1d 6665 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (((#‘𝐴) + (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) − (#‘𝐴)) = ((#‘𝐵) − (#‘𝐴))) |
181 | 146, 180 | eqtr3d 2658 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑎:𝐴–1-1→𝐵) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((#‘𝐵) − (#‘𝐴))) |
182 | 140, 181 | sylan2 491 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((#‘𝐵) − (#‘𝐴))) |
183 | 182 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝑎 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})) = ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + ((#‘𝐵) − (#‘𝐴)))) |
184 | 132, 183 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + ((#‘𝐵) − (#‘𝐴)))) |
185 | | hashunsng 13181 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (#‘(𝑦 ∪ {𝑎})) = ((#‘𝑦) + 1))) |
186 | 135, 185 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (#‘(𝑦 ∪ {𝑎})) = ((#‘𝑦) + 1)) |
187 | 186 | ad2antrl 764 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (#‘(𝑦 ∪ {𝑎})) = ((#‘𝑦) + 1)) |
188 | 187 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎}))) = (((#‘𝐵) − (#‘𝐴)) · ((#‘𝑦) + 1))) |
189 | 80 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((#‘𝐵) − (#‘𝐴)) ∈ ℂ) |
190 | | simprll 802 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → 𝑦 ∈ Fin) |
191 | | hashcl 13147 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ Fin →
(#‘𝑦) ∈
ℕ0) |
192 | 190, 191 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (#‘𝑦) ∈
ℕ0) |
193 | 192 | nn0cnd 11353 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (#‘𝑦) ∈ ℂ) |
194 | | 1cnd 10056 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → 1 ∈
ℂ) |
195 | 189, 193,
194 | adddid 10064 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((#‘𝐵) − (#‘𝐴)) · ((#‘𝑦) + 1)) = ((((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)) + (((#‘𝐵) − (#‘𝐴)) · 1))) |
196 | 189 | mulid1d 10057 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((#‘𝐵) − (#‘𝐴)) · 1) = ((#‘𝐵) − (#‘𝐴))) |
197 | 196 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)) + (((#‘𝐵) − (#‘𝐴)) · 1)) = ((((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)) + ((#‘𝐵) − (#‘𝐴)))) |
198 | 188, 195,
197 | 3eqtrd 2660 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎}))) = ((((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)) + ((#‘𝐵) − (#‘𝐴)))) |
199 | 184, 198 | eqeq12d 2637 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎}))) ↔ ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) + ((#‘𝐵) − (#‘𝐴))) = ((((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)) + ((#‘𝐵) − (#‘𝐴))))) |
200 | 88, 199 | syl5ibr 236 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) ∧ (𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵})) → ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎}))))) |
201 | 200 | expr 643 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦)) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → ((#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)) → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎})))))) |
202 | 201 | a2d 29 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦)) → (((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦))) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎})))))) |
203 | 87, 202 | syl5 34 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦)) → ((𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦))) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎})))))) |
204 | 203 | expcom 451 |
. . . . 5
⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → (𝜑 → ((𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦))) → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎}))))))) |
205 | 204 | a2d 29 |
. . . 4
⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑦) → ((𝜑 → (𝑦 ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ 𝑦 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘𝑦)))) → (𝜑 → ((𝑦 ∪ {𝑎}) ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ ((𝑓 ↾ 𝐴) ∈ (𝑦 ∪ {𝑎}) ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}) = (((#‘𝐵) − (#‘𝐴)) · (#‘(𝑦 ∪ {𝑎}))))))) |
206 | 29, 39, 49, 73, 83, 205 | findcard2s 8201 |
. . 3
⊢ ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ Fin → (𝜑 → ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))))) |
207 | 11, 206 | mpcom 38 |
. 2
⊢ (𝜑 → ({𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} → (#‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵})))) |
208 | 1, 207 | mpi 20 |
1
⊢ (𝜑 → (#‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))) |