Step |
Hyp |
Ref |
Expression |
1 |
|
disjf1o.xph |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
disjf1o.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
3 |
|
disjf1o.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
4 |
|
disjf1o.dj |
⊢ ( 𝜑 → Disj 𝑥 ∈ 𝐴 𝐵 ) |
5 |
|
disjf1o.d |
⊢ 𝐶 = { 𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } |
6 |
|
disjf1o.e |
⊢ 𝐷 = ( ran 𝐹 ∖ { ∅ } ) |
7 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) |
8 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝜑 ) |
9 |
|
ssrab2 |
⊢ { 𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ⊆ 𝐴 |
10 |
5 9
|
eqsstri |
⊢ 𝐶 ⊆ 𝐴 |
11 |
|
id |
⊢ ( 𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐶 ) |
12 |
10 11
|
sseldi |
⊢ ( 𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴 ) |
13 |
12
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝑥 ∈ 𝐴 ) |
14 |
8 13 3
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝐵 ∈ 𝑉 ) |
15 |
11 5
|
eleqtrdi |
⊢ ( 𝑥 ∈ 𝐶 → 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ) |
16 |
|
rabid |
⊢ ( 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ↔ ( 𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ) ) |
17 |
16
|
a1i |
⊢ ( 𝑥 ∈ 𝐶 → ( 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ↔ ( 𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ) ) ) |
18 |
15 17
|
mpbid |
⊢ ( 𝑥 ∈ 𝐶 → ( 𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ) ) |
19 |
18
|
simprd |
⊢ ( 𝑥 ∈ 𝐶 → 𝐵 ≠ ∅ ) |
20 |
19
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝐵 ≠ ∅ ) |
21 |
10
|
a1i |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 ) |
22 |
|
disjss1 |
⊢ ( 𝐶 ⊆ 𝐴 → ( Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐶 𝐵 ) ) |
23 |
21 4 22
|
sylc |
⊢ ( 𝜑 → Disj 𝑥 ∈ 𝐶 𝐵 ) |
24 |
1 7 14 20 23
|
disjf1 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) : 𝐶 –1-1→ 𝑉 ) |
25 |
|
f1f1orn |
⊢ ( ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) : 𝐶 –1-1→ 𝑉 → ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) : 𝐶 –1-1-onto→ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
26 |
24 25
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) : 𝐶 –1-1-onto→ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
27 |
2
|
a1i |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
28 |
27
|
reseq1d |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐶 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) ) |
29 |
21
|
resmptd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
30 |
28 29
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐶 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
31 |
|
eqidd |
⊢ ( 𝜑 → 𝐶 = 𝐶 ) |
32 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝜑 ) |
33 |
|
id |
⊢ ( 𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐷 ) |
34 |
33 6
|
eleqtrdi |
⊢ ( 𝑦 ∈ 𝐷 → 𝑦 ∈ ( ran 𝐹 ∖ { ∅ } ) ) |
35 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( ran 𝐹 ∖ { ∅ } ) → 𝑦 ≠ ∅ ) |
36 |
34 35
|
syl |
⊢ ( 𝑦 ∈ 𝐷 → 𝑦 ≠ ∅ ) |
37 |
36
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ≠ ∅ ) |
38 |
|
eldifi |
⊢ ( 𝑦 ∈ ( ran 𝐹 ∖ { ∅ } ) → 𝑦 ∈ ran 𝐹 ) |
39 |
34 38
|
syl |
⊢ ( 𝑦 ∈ 𝐷 → 𝑦 ∈ ran 𝐹 ) |
40 |
2
|
elrnmpt |
⊢ ( 𝑦 ∈ ran 𝐹 → ( 𝑦 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) ) |
41 |
39 40
|
syl |
⊢ ( 𝑦 ∈ 𝐷 → ( 𝑦 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) ) |
42 |
39 41
|
mpbid |
⊢ ( 𝑦 ∈ 𝐷 → ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) |
43 |
42
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) |
44 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 ≠ ∅ |
45 |
1 44
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑦 ≠ ∅ ) |
46 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑦 |
47 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) |
48 |
47
|
nfrn |
⊢ Ⅎ 𝑥 ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) |
49 |
46 48
|
nfel |
⊢ Ⅎ 𝑥 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) |
50 |
|
simp3 |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 ) |
51 |
|
simp2 |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑥 ∈ 𝐴 ) |
52 |
|
id |
⊢ ( 𝑦 = 𝐵 → 𝑦 = 𝐵 ) |
53 |
52
|
eqcomd |
⊢ ( 𝑦 = 𝐵 → 𝐵 = 𝑦 ) |
54 |
53
|
adantl |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑦 = 𝐵 ) → 𝐵 = 𝑦 ) |
55 |
|
simpl |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑦 = 𝐵 ) → 𝑦 ≠ ∅ ) |
56 |
54 55
|
eqnetrd |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑦 = 𝐵 ) → 𝐵 ≠ ∅ ) |
57 |
56
|
3adant2 |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝐵 ≠ ∅ ) |
58 |
51 57
|
jca |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑥 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ) ) |
59 |
58 16
|
sylibr |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } ) |
60 |
5
|
eqcomi |
⊢ { 𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } = 𝐶 |
61 |
60
|
a1i |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → { 𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅ } = 𝐶 ) |
62 |
59 61
|
eleqtrd |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑥 ∈ 𝐶 ) |
63 |
|
eqvisset |
⊢ ( 𝑦 = 𝐵 → 𝐵 ∈ V ) |
64 |
63
|
3ad2ant3 |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝐵 ∈ V ) |
65 |
7
|
elrnmpt1 |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝐵 ∈ V ) → 𝐵 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
66 |
62 64 65
|
syl2anc |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝐵 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
67 |
50 66
|
eqeltrd |
⊢ ( ( 𝑦 ≠ ∅ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
68 |
67
|
3adant1l |
⊢ ( ( ( 𝜑 ∧ 𝑦 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
69 |
68
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑦 ≠ ∅ ) → ( 𝑥 ∈ 𝐴 → ( 𝑦 = 𝐵 → 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) ) ) |
70 |
45 49 69
|
rexlimd |
⊢ ( ( 𝜑 ∧ 𝑦 ≠ ∅ ) → ( ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) ) |
71 |
70
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑦 ≠ ∅ ) ∧ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) → 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
72 |
32 37 43 71
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
73 |
72
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐷 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
74 |
|
dfss3 |
⊢ ( 𝐷 ⊆ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ↔ ∀ 𝑦 ∈ 𝐷 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
75 |
73 74
|
sylibr |
⊢ ( 𝜑 → 𝐷 ⊆ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
76 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) → 𝜑 ) |
77 |
|
vex |
⊢ 𝑦 ∈ V |
78 |
7
|
elrnmpt |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐶 𝑦 = 𝐵 ) ) |
79 |
77 78
|
ax-mp |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐶 𝑦 = 𝐵 ) |
80 |
79
|
biimpi |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) → ∃ 𝑥 ∈ 𝐶 𝑦 = 𝐵 ) |
81 |
80
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) → ∃ 𝑥 ∈ 𝐶 𝑦 = 𝐵 ) |
82 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 ∈ 𝐷 |
83 |
|
simpr |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 ) |
84 |
12
|
adantr |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝑥 ∈ 𝐴 ) |
85 |
83 63
|
syl |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝐵 ∈ V ) |
86 |
2
|
elrnmpt1 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V ) → 𝐵 ∈ ran 𝐹 ) |
87 |
84 85 86
|
syl2anc |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝐵 ∈ ran 𝐹 ) |
88 |
83 87
|
eqeltrd |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ ran 𝐹 ) |
89 |
88
|
3adant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ ran 𝐹 ) |
90 |
19
|
adantr |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝐵 ≠ ∅ ) |
91 |
83 90
|
eqnetrd |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝑦 ≠ ∅ ) |
92 |
|
nelsn |
⊢ ( 𝑦 ≠ ∅ → ¬ 𝑦 ∈ { ∅ } ) |
93 |
91 92
|
syl |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → ¬ 𝑦 ∈ { ∅ } ) |
94 |
93
|
3adant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → ¬ 𝑦 ∈ { ∅ } ) |
95 |
89 94
|
eldifd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ ( ran 𝐹 ∖ { ∅ } ) ) |
96 |
95 6
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ 𝐷 ) |
97 |
96
|
3exp |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐶 → ( 𝑦 = 𝐵 → 𝑦 ∈ 𝐷 ) ) ) |
98 |
1 82 97
|
rexlimd |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐶 𝑦 = 𝐵 → 𝑦 ∈ 𝐷 ) ) |
99 |
98
|
imp |
⊢ ( ( 𝜑 ∧ ∃ 𝑥 ∈ 𝐶 𝑦 = 𝐵 ) → 𝑦 ∈ 𝐷 ) |
100 |
76 81 99
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) → 𝑦 ∈ 𝐷 ) |
101 |
75 100
|
eqelssd |
⊢ ( 𝜑 → 𝐷 = ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
102 |
30 31 101
|
f1oeq123d |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ 𝐷 ↔ ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) : 𝐶 –1-1-onto→ ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) ) |
103 |
26 102
|
mpbird |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ 𝐷 ) |