Step | Hyp | Ref
| Expression |
1 | | ssmapsn.c |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ⊆ (𝐵 ↑m {𝐴})) |
2 | 1 | sselda 3994 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐵 ↑m {𝐴})) |
3 | | elmapi 8887 |
. . . . . . 7
⊢ (𝑓 ∈ (𝐵 ↑m {𝐴}) → 𝑓:{𝐴}⟶𝐵) |
4 | 2, 3 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓:{𝐴}⟶𝐵) |
5 | 4 | ffnd 6737 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 Fn {𝐴}) |
6 | | ssmapsn.d |
. . . . . . . 8
⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓 |
7 | 6 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓) |
8 | | ovexd 7465 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵 ↑m {𝐴}) ∈ V) |
9 | 8, 1 | ssexd 5329 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ V) |
10 | | rnexg 7924 |
. . . . . . . . 9
⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ∈ V) |
11 | 10 | rgen 3060 |
. . . . . . . 8
⊢
∀𝑓 ∈
𝐶 ran 𝑓 ∈ V |
12 | | iunexg 7986 |
. . . . . . . 8
⊢ ((𝐶 ∈ V ∧ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V) |
13 | 9, 11, 12 | sylancl 586 |
. . . . . . 7
⊢ (𝜑 → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V) |
14 | 7, 13 | eqeltrd 2838 |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ V) |
15 | 14 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐷 ∈ V) |
16 | | ssiun2 5051 |
. . . . . . . 8
⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ⊆ ∪
𝑓 ∈ 𝐶 ran 𝑓) |
17 | 16 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → ran 𝑓 ⊆ ∪
𝑓 ∈ 𝐶 ran 𝑓) |
18 | | ssmapsn.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
19 | | snidg 4664 |
. . . . . . . . . 10
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) |
20 | 18, 19 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ {𝐴}) |
21 | 20 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐴 ∈ {𝐴}) |
22 | 5, 21 | fnfvelrnd 7101 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ran 𝑓) |
23 | 17, 22 | sseldd 3995 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ∪
𝑓 ∈ 𝐶 ran 𝑓) |
24 | 23, 6 | eleqtrrdi 2849 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ 𝐷) |
25 | 5, 15, 24 | elmapsnd 45146 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐷 ↑m {𝐴})) |
26 | 14 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐷 ∈ V) |
27 | | snex 5441 |
. . . . . . . . 9
⊢ {𝐴} ∈ V |
28 | 27 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → {𝐴} ∈ V) |
29 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝑓 ∈ (𝐷 ↑m {𝐴})) |
30 | 20 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐴 ∈ {𝐴}) |
31 | 26, 28, 29, 30 | fvmap 45140 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ 𝐷) |
32 | | rneq 5949 |
. . . . . . . . 9
⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔) |
33 | 32 | cbviunv 5044 |
. . . . . . . 8
⊢ ∪ 𝑓 ∈ 𝐶 ran 𝑓 = ∪ 𝑔 ∈ 𝐶 ran 𝑔 |
34 | 6, 33 | eqtri 2762 |
. . . . . . 7
⊢ 𝐷 = ∪ 𝑔 ∈ 𝐶 ran 𝑔 |
35 | 31, 34 | eleqtrdi 2848 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ ∪
𝑔 ∈ 𝐶 ran 𝑔) |
36 | | eliun 4999 |
. . . . . 6
⊢ ((𝑓‘𝐴) ∈ ∪
𝑔 ∈ 𝐶 ran 𝑔 ↔ ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔) |
37 | 35, 36 | sylib 218 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔) |
38 | | simp3 1137 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓‘𝐴) ∈ ran 𝑔) |
39 | | simp1l 1196 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝜑) |
40 | 39, 18 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝐴 ∈ 𝑉) |
41 | | eqid 2734 |
. . . . . . . . 9
⊢ {𝐴} = {𝐴} |
42 | | simp1r 1197 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷 ↑m {𝐴})) |
43 | | elmapfn 8903 |
. . . . . . . . . 10
⊢ (𝑓 ∈ (𝐷 ↑m {𝐴}) → 𝑓 Fn {𝐴}) |
44 | 42, 43 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴}) |
45 | 1 | sselda 3994 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ (𝐵 ↑m {𝐴})) |
46 | | elmapfn 8903 |
. . . . . . . . . . . 12
⊢ (𝑔 ∈ (𝐵 ↑m {𝐴}) → 𝑔 Fn {𝐴}) |
47 | 45, 46 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 Fn {𝐴}) |
48 | 47 | 3adant3 1131 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴}) |
49 | 48 | 3adant1r 1176 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴}) |
50 | 40, 41, 44, 49 | fsneqrn 45153 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓‘𝐴) ∈ ran 𝑔)) |
51 | 38, 50 | mpbird 257 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔) |
52 | | simp2 1136 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 ∈ 𝐶) |
53 | 51, 52 | eqeltrd 2838 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ 𝐶) |
54 | 53 | rexlimdv3a 3156 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔 → 𝑓 ∈ 𝐶)) |
55 | 37, 54 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝑓 ∈ 𝐶) |
56 | 25, 55 | impbida 801 |
. . 3
⊢ (𝜑 → (𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴}))) |
57 | 56 | alrimiv 1924 |
. 2
⊢ (𝜑 → ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴}))) |
58 | | nfcv 2902 |
. . 3
⊢
Ⅎ𝑓𝐶 |
59 | | ssmapsn.f |
. . . 4
⊢
Ⅎ𝑓𝐷 |
60 | | nfcv 2902 |
. . . 4
⊢
Ⅎ𝑓
↑m |
61 | | nfcv 2902 |
. . . 4
⊢
Ⅎ𝑓{𝐴} |
62 | 59, 60, 61 | nfov 7460 |
. . 3
⊢
Ⅎ𝑓(𝐷 ↑m {𝐴}) |
63 | 58, 62 | cleqf 2931 |
. 2
⊢ (𝐶 = (𝐷 ↑m {𝐴}) ↔ ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴}))) |
64 | 57, 63 | sylibr 234 |
1
⊢ (𝜑 → 𝐶 = (𝐷 ↑m {𝐴})) |