Step |
Hyp |
Ref |
Expression |
1 |
|
choicefi.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
2 |
|
choicefi.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑊 ) |
3 |
|
choicefi.n |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≠ ∅ ) |
4 |
|
mptfi |
⊢ ( 𝐴 ∈ Fin → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ Fin ) |
5 |
|
rnfi |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ Fin → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ Fin ) |
6 |
|
fnchoice |
⊢ ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ Fin → ∃ 𝑔 ( 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) ) |
7 |
1 4 5 6
|
4syl |
⊢ ( 𝜑 → ∃ 𝑔 ( 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) ) |
8 |
|
simpl |
⊢ ( ( 𝜑 ∧ ( 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) ) → 𝜑 ) |
9 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) ) → 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
10 |
|
nfv |
⊢ Ⅎ 𝑦 𝜑 |
11 |
|
nfra1 |
⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) |
12 |
10 11
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) |
13 |
|
rspa |
⊢ ( ( ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) |
14 |
13
|
adantll |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) |
15 |
|
vex |
⊢ 𝑦 ∈ V |
16 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
17 |
16
|
elrnmpt |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) ) |
18 |
15 17
|
ax-mp |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) |
19 |
18
|
biimpi |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) |
20 |
19
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) |
21 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 ) |
22 |
3
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝐵 ≠ ∅ ) |
23 |
21 22
|
eqnetrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑦 ≠ ∅ ) |
24 |
23
|
3exp |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐵 → 𝑦 ≠ ∅ ) ) ) |
25 |
24
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ≠ ∅ ) ) |
26 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ≠ ∅ ) ) |
27 |
20 26
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → 𝑦 ≠ ∅ ) |
28 |
27
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → 𝑦 ≠ ∅ ) |
29 |
|
id |
⊢ ( ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) → ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) |
30 |
29
|
imp |
⊢ ( ( ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑦 ≠ ∅ ) → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) |
31 |
14 28 30
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) |
32 |
31
|
ex |
⊢ ( ( 𝜑 ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) → ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) |
33 |
12 32
|
ralrimi |
⊢ ( ( 𝜑 ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) → ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) |
34 |
|
rsp |
⊢ ( ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 → ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) |
35 |
33 34
|
syl |
⊢ ( ( 𝜑 ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) → ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) |
36 |
12 35
|
ralrimi |
⊢ ( ( 𝜑 ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) → ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) |
37 |
36
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑦 ≠ ∅ → ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ) ) → ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) |
38 |
|
vex |
⊢ 𝑔 ∈ V |
39 |
38
|
a1i |
⊢ ( 𝜑 → 𝑔 ∈ V ) |
40 |
1
|
mptexd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ V ) |
41 |
|
coexg |
⊢ ( ( 𝑔 ∈ V ∧ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ V ) → ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ V ) |
42 |
39 40 41
|
syl2anc |
⊢ ( 𝜑 → ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ V ) |
43 |
42
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) → ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ V ) |
44 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
45 |
2
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ) |
46 |
16
|
fnmpt |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 ) |
47 |
45 46
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 ) |
48 |
47
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 ) |
49 |
|
ssidd |
⊢ ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
50 |
|
fnco |
⊢ ( ( 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 ) |
51 |
44 48 49 50
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) Fn 𝐴 ) |
52 |
51
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) → ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) Fn 𝐴 ) |
53 |
|
nfv |
⊢ Ⅎ 𝑥 𝜑 |
54 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑔 |
55 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
56 |
55
|
nfrn |
⊢ Ⅎ 𝑥 ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
57 |
54 56
|
nffn |
⊢ Ⅎ 𝑥 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
58 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 |
59 |
56 58
|
nfralw |
⊢ Ⅎ 𝑥 ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 |
60 |
53 57 59
|
nf3an |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) |
61 |
|
funmpt |
⊢ Fun ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
62 |
61
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → Fun ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
63 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
64 |
16 2
|
dmmptd |
⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
65 |
64
|
eqcomd |
⊢ ( 𝜑 → 𝐴 = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
66 |
65
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐴 = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
67 |
63 66
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
68 |
|
fvco |
⊢ ( ( Fun ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ 𝑥 ∈ dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) = ( 𝑔 ‘ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ) ) |
69 |
62 67 68
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) = ( 𝑔 ‘ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ) ) |
70 |
16
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 ) |
71 |
63 2 70
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 ) |
72 |
71
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑔 ‘ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ) = ( 𝑔 ‘ 𝐵 ) ) |
73 |
69 72
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) = ( 𝑔 ‘ 𝐵 ) ) |
74 |
73
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) = ( 𝑔 ‘ 𝐵 ) ) |
75 |
16
|
elrnmpt1 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊 ) → 𝐵 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
76 |
63 2 75
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
77 |
76
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
78 |
|
simpl3 |
⊢ ( ( ( 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 ) |
79 |
|
fveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝑔 ‘ 𝑦 ) = ( 𝑔 ‘ 𝐵 ) ) |
80 |
|
id |
⊢ ( 𝑦 = 𝐵 → 𝑦 = 𝐵 ) |
81 |
79 80
|
eleq12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ↔ ( 𝑔 ‘ 𝐵 ) ∈ 𝐵 ) ) |
82 |
81
|
rspcva |
⊢ ( ( 𝐵 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) → ( 𝑔 ‘ 𝐵 ) ∈ 𝐵 ) |
83 |
77 78 82
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑔 ‘ 𝐵 ) ∈ 𝐵 ) |
84 |
74 83
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) ∈ 𝐵 ) |
85 |
84
|
ex |
⊢ ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) → ( 𝑥 ∈ 𝐴 → ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) ∈ 𝐵 ) ) |
86 |
60 85
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) → ∀ 𝑥 ∈ 𝐴 ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) ∈ 𝐵 ) |
87 |
52 86
|
jca |
⊢ ( ( 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 ) ) |
88 |
|
fneq1 |
⊢ ( 𝑓 = ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( 𝑓 Fn 𝐴 ↔ ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) Fn 𝐴 ) ) |
89 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑓 |
90 |
54 55
|
nfco |
⊢ Ⅎ 𝑥 ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
91 |
89 90
|
nfeq |
⊢ Ⅎ 𝑥 𝑓 = ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
92 |
|
fveq1 |
⊢ ( 𝑓 = ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( 𝑓 ‘ 𝑥 ) = ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) ) |
93 |
92
|
eleq1d |
⊢ ( 𝑓 = ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ↔ ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) ∈ 𝐵 ) ) |
94 |
91 93
|
ralbid |
⊢ ( 𝑓 = ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ( ( 𝑔 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ‘ 𝑥 ) ∈ 𝐵 ) ) |
95 |
88 94
|
anbi12d |
⊢ ( 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 ) ) ) |
96 |
95
|
spcegv |
⊢ ( ( 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 ) ) ) |
97 |
43 87 96
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑔 Fn ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ∀ 𝑦 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ( 𝑔 ‘ 𝑦 ) ∈ 𝑦 ) → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) |
98 |
8 9 37 97
|
syl3anc |
⊢ ( ( 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 ) ) |
99 |
98
|
ex |
⊢ ( 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 ) ) ) |
100 |
99
|
exlimdv |
⊢ ( 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 ) ) ) |
101 |
7 100
|
mpd |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) |