Step | Hyp | Ref
| Expression |
1 | | choicefi.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ Fin) |
2 | | mptfi 9118 |
. . . . 5
⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) |
4 | | rnfi 9102 |
. . . 4
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) |
5 | 3, 4 | syl 17 |
. . 3
⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) |
6 | | fnchoice 42572 |
. . 3
⊢ (ran
(𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin → ∃𝑔(𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) |
7 | 5, 6 | syl 17 |
. 2
⊢ (𝜑 → ∃𝑔(𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) |
8 | | simpl 483 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → 𝜑) |
9 | | simprl 768 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
10 | | nfv 1917 |
. . . . . . . 8
⊢
Ⅎ𝑦𝜑 |
11 | | nfra1 3144 |
. . . . . . . 8
⊢
Ⅎ𝑦∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) |
12 | 10, 11 | nfan 1902 |
. . . . . . 7
⊢
Ⅎ𝑦(𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) |
13 | | rspa 3132 |
. . . . . . . . . . . 12
⊢
((∀𝑦 ∈
ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) |
14 | 13 | adantll 711 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) |
15 | | vex 3436 |
. . . . . . . . . . . . . . . 16
⊢ 𝑦 ∈ V |
16 | | eqid 2738 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
17 | 16 | elrnmpt 5865 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)) |
18 | 15, 17 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵) |
19 | 18 | biimpi 215 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵) |
20 | 19 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵) |
21 | | simp3 1137 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) |
22 | | choicefi.n |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅) |
23 | 22 | 3adant3 1131 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅) |
24 | 21, 23 | eqnetrd 3011 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅) |
25 | 24 | 3exp 1118 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐵 → 𝑦 ≠ ∅))) |
26 | 25 | rexlimdv 3212 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ≠ ∅)) |
27 | 26 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ≠ ∅)) |
28 | 20, 27 | mpd 15 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ≠ ∅) |
29 | 28 | adantlr 712 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ≠ ∅) |
30 | | id 22 |
. . . . . . . . . . . 12
⊢ ((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → (𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) |
31 | 30 | imp 407 |
. . . . . . . . . . 11
⊢ (((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ 𝑦 ≠ ∅) → (𝑔‘𝑦) ∈ 𝑦) |
32 | 14, 29, 31 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑔‘𝑦) ∈ 𝑦) |
33 | 32 | ex 413 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑦) ∈ 𝑦)) |
34 | 12, 33 | ralrimi 3141 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) |
35 | | rsp 3131 |
. . . . . . . 8
⊢
(∀𝑦 ∈
ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑦) ∈ 𝑦)) |
36 | 34, 35 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑦) ∈ 𝑦)) |
37 | 12, 36 | ralrimi 3141 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) |
38 | 37 | adantrl 713 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) |
39 | | vex 3436 |
. . . . . . . . 9
⊢ 𝑔 ∈ V |
40 | 39 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝑔 ∈ V) |
41 | 1 | mptexd 7100 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
42 | | coexg 7776 |
. . . . . . . 8
⊢ ((𝑔 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V) |
43 | 40, 41, 42 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V) |
44 | 43 | 3ad2ant1 1132 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V) |
45 | | simpr 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
46 | | choicefi.b |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
47 | 46 | ralrimiva 3103 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) |
48 | 16 | fnmpt 6573 |
. . . . . . . . . . 11
⊢
(∀𝑥 ∈
𝐴 𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
49 | 47, 48 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
50 | 49 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
51 | | ssidd 3944 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
52 | | fnco 6549 |
. . . . . . . . 9
⊢ ((𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴) |
53 | 45, 50, 51, 52 | syl3anc 1370 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴) |
54 | 53 | 3adant3 1131 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴) |
55 | | nfv 1917 |
. . . . . . . . 9
⊢
Ⅎ𝑥𝜑 |
56 | | nfcv 2907 |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝑔 |
57 | | nfmpt1 5182 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) |
58 | 57 | nfrn 5861 |
. . . . . . . . . 10
⊢
Ⅎ𝑥ran
(𝑥 ∈ 𝐴 ↦ 𝐵) |
59 | 56, 58 | nffn 6532 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
60 | | nfv 1917 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑔‘𝑦) ∈ 𝑦 |
61 | 58, 60 | nfralw 3151 |
. . . . . . . . 9
⊢
Ⅎ𝑥∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦 |
62 | 55, 59, 61 | nf3an 1904 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) |
63 | | funmpt 6472 |
. . . . . . . . . . . . . 14
⊢ Fun
(𝑥 ∈ 𝐴 ↦ 𝐵) |
64 | 63 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Fun (𝑥 ∈ 𝐴 ↦ 𝐵)) |
65 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) |
66 | 16, 46 | dmmptd 6578 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
67 | 66 | eqcomd 2744 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
68 | 67 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
69 | 65, 68 | eleqtrd 2841 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
70 | | fvco 6866 |
. . . . . . . . . . . . 13
⊢ ((Fun
(𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵)) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) |
71 | 64, 69, 70 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) |
72 | 16 | fvmpt2 6886 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
73 | 65, 46, 72 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
74 | 73 | fveq2d 6778 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑔‘((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑔‘𝐵)) |
75 | 71, 74 | eqtrd 2778 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘𝐵)) |
76 | 75 | 3ad2antl1 1184 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘𝐵)) |
77 | 16 | elrnmpt1 5867 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
78 | 65, 46, 77 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
79 | 78 | 3ad2antl1 1184 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
80 | | simpl3 1192 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) |
81 | | fveq2 6774 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝐵 → (𝑔‘𝑦) = (𝑔‘𝐵)) |
82 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) |
83 | 81, 82 | eleq12d 2833 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝐵 → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘𝐵) ∈ 𝐵)) |
84 | 83 | rspcva 3559 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑔‘𝐵) ∈ 𝐵) |
85 | 79, 80, 84 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝐵) ∈ 𝐵) |
86 | 76, 85 | eqeltrd 2839 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵) |
87 | 86 | ex 413 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑥 ∈ 𝐴 → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵)) |
88 | 62, 87 | ralrimi 3141 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵) |
89 | 54, 88 | jca 512 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵)) |
90 | | fneq1 6524 |
. . . . . . . 8
⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑓 Fn 𝐴 ↔ (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴)) |
91 | | nfcv 2907 |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝑓 |
92 | 56, 57 | nfco 5774 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) |
93 | 91, 92 | nfeq 2920 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) |
94 | | fveq1 6773 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑓‘𝑥) = ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥)) |
95 | 94 | eleq1d 2823 |
. . . . . . . . 9
⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → ((𝑓‘𝑥) ∈ 𝐵 ↔ ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵)) |
96 | 93, 95 | ralbid 3161 |
. . . . . . . 8
⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵)) |
97 | 90, 96 | anbi12d 631 |
. . . . . . 7
⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵))) |
98 | 97 | spcegv 3536 |
. . . . . 6
⊢ ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V → (((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))) |
99 | 44, 89, 98 | sylc 65 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
100 | 8, 9, 38, 99 | syl3anc 1370 |
. . . 4
⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
101 | 100 | ex 413 |
. . 3
⊢ (𝜑 → ((𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))) |
102 | 101 | exlimdv 1936 |
. 2
⊢ (𝜑 → (∃𝑔(𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))) |
103 | 7, 102 | mpd 15 |
1
⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |