Step | Hyp | Ref
| Expression |
1 | | unirnmapsn.C |
. . . . 5
⊢ 𝐶 = {𝐴} |
2 | | snex 5057 |
. . . . 5
⊢ {𝐴} ∈ V |
3 | 1, 2 | eqeltri 2835 |
. . . 4
⊢ 𝐶 ∈ V |
4 | 3 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐶 ∈ V) |
5 | | unirnmapsn.x |
. . 3
⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 𝐶)) |
6 | 4, 5 | unirnmap 39917 |
. 2
⊢ (𝜑 → 𝑋 ⊆ (ran ∪
𝑋
↑𝑚 𝐶)) |
7 | | simpl 474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪
𝑋
↑𝑚 𝐶)) → 𝜑) |
8 | | equid 2094 |
. . . . . . . . 9
⊢ 𝑔 = 𝑔 |
9 | | rnuni 5702 |
. . . . . . . . . 10
⊢ ran ∪ 𝑋 =
∪ 𝑓 ∈ 𝑋 ran 𝑓 |
10 | 9 | oveq1i 6824 |
. . . . . . . . 9
⊢ (ran
∪ 𝑋 ↑𝑚 𝐶) = (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶) |
11 | 8, 10 | eleq12i 2832 |
. . . . . . . 8
⊢ (𝑔 ∈ (ran ∪ 𝑋
↑𝑚 𝐶) ↔ 𝑔 ∈ (∪
𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) |
12 | 11 | biimpi 206 |
. . . . . . 7
⊢ (𝑔 ∈ (ran ∪ 𝑋
↑𝑚 𝐶) → 𝑔 ∈ (∪
𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) |
13 | 12 | adantl 473 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪
𝑋
↑𝑚 𝐶)) → 𝑔 ∈ (∪
𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) |
14 | | ovexd 6844 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵 ↑𝑚 𝐶) ∈ V) |
15 | 14, 5 | ssexd 4957 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈ V) |
16 | | rnexg 7264 |
. . . . . . . . . . . . 13
⊢ (𝑓 ∈ 𝑋 → ran 𝑓 ∈ V) |
17 | 16 | rgen 3060 |
. . . . . . . . . . . 12
⊢
∀𝑓 ∈
𝑋 ran 𝑓 ∈ V |
18 | 17 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V) |
19 | | iunexg 7309 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ V ∧ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V) |
20 | 15, 18, 19 | syl2anc 696 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V) |
21 | 20, 4 | elmapd 8039 |
. . . . . . . . 9
⊢ (𝜑 → (𝑔 ∈ (∪
𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶) ↔ 𝑔:𝐶⟶∪
𝑓 ∈ 𝑋 ran 𝑓)) |
22 | 21 | biimpa 502 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ (∪
𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝑔:𝐶⟶∪
𝑓 ∈ 𝑋 ran 𝑓) |
23 | | unirnmapsn.A |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
24 | | snidg 4351 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) |
25 | 23, 24 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ {𝐴}) |
26 | 25, 1 | syl6eleqr 2850 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ 𝐶) |
27 | 26 | adantr 472 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ (∪
𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝐴 ∈ 𝐶) |
28 | 22, 27 | ffvelrnd 6524 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ (∪
𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → (𝑔‘𝐴) ∈ ∪
𝑓 ∈ 𝑋 ran 𝑓) |
29 | | eliun 4676 |
. . . . . . 7
⊢ ((𝑔‘𝐴) ∈ ∪
𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓) |
30 | 28, 29 | sylib 208 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ (∪
𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓) |
31 | 7, 13, 30 | syl2anc 696 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪
𝑋
↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓) |
32 | | elmapfn 8048 |
. . . . . . . 8
⊢ (𝑔 ∈ (ran ∪ 𝑋
↑𝑚 𝐶) → 𝑔 Fn 𝐶) |
33 | 32 | adantl 473 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪
𝑋
↑𝑚 𝐶)) → 𝑔 Fn 𝐶) |
34 | | simp3 1133 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ ran 𝑓) |
35 | 23 | 3ad2ant1 1128 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉) |
36 | 1 | oveq2i 6825 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐵 ↑𝑚
𝐶) = (𝐵 ↑𝑚 {𝐴}) |
37 | 5, 36 | syl6sseq 3792 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴})) |
38 | 37 | adantr 472 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴})) |
39 | | simpr 479 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋) |
40 | 38, 39 | sseldd 3745 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 {𝐴})) |
41 | | unirnmapsn.b |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐵 ∈ 𝑊) |
42 | 41 | adantr 472 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐵 ∈ 𝑊) |
43 | 2 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝐴} ∈ V) |
44 | 42, 43 | elmapd 8039 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ∈ (𝐵 ↑𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵)) |
45 | 40, 44 | mpbid 222 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶𝐵) |
46 | 45 | 3adant3 1127 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵) |
47 | 35, 46 | rnsnf 39887 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓‘𝐴)}) |
48 | 34, 47 | eleqtrd 2841 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ {(𝑓‘𝐴)}) |
49 | | fvex 6363 |
. . . . . . . . . . . . 13
⊢ (𝑔‘𝐴) ∈ V |
50 | 49 | elsn 4336 |
. . . . . . . . . . . 12
⊢ ((𝑔‘𝐴) ∈ {(𝑓‘𝐴)} ↔ (𝑔‘𝐴) = (𝑓‘𝐴)) |
51 | 48, 50 | sylib 208 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴)) |
52 | 51 | 3adant1r 1188 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴)) |
53 | 23 | adantr 472 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → 𝐴 ∈ 𝑉) |
54 | 53 | 3ad2ant1 1128 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉) |
55 | | simp1r 1241 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶) |
56 | 40, 36 | syl6eleqr 2850 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 𝐶)) |
57 | | elmapfn 8048 |
. . . . . . . . . . . . . 14
⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐶) → 𝑓 Fn 𝐶) |
58 | 56, 57 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶) |
59 | 58 | adantlr 753 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶) |
60 | 59 | 3adant3 1127 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶) |
61 | 54, 1, 55, 60 | fsneq 39915 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔‘𝐴) = (𝑓‘𝐴))) |
62 | 52, 61 | mpbird 247 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓) |
63 | | simp2 1132 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 ∈ 𝑋) |
64 | 62, 63 | eqeltrd 2839 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 ∈ 𝑋) |
65 | 64 | 3exp 1113 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋))) |
66 | 7, 33, 65 | syl2anc 696 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪
𝑋
↑𝑚 𝐶)) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋))) |
67 | 66 | rexlimdv 3168 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪
𝑋
↑𝑚 𝐶)) → (∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)) |
68 | 31, 67 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪
𝑋
↑𝑚 𝐶)) → 𝑔 ∈ 𝑋) |
69 | 68 | ralrimiva 3104 |
. . 3
⊢ (𝜑 → ∀𝑔 ∈ (ran ∪
𝑋
↑𝑚 𝐶)𝑔 ∈ 𝑋) |
70 | | dfss3 3733 |
. . 3
⊢ ((ran
∪ 𝑋 ↑𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran ∪
𝑋
↑𝑚 𝐶)𝑔 ∈ 𝑋) |
71 | 69, 70 | sylibr 224 |
. 2
⊢ (𝜑 → (ran ∪ 𝑋
↑𝑚 𝐶) ⊆ 𝑋) |
72 | 6, 71 | eqssd 3761 |
1
⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑𝑚
𝐶)) |