Step |
Hyp |
Ref |
Expression |
1 |
|
disjf1.xph |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
disjf1.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
3 |
|
disjf1.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
4 |
|
disjf1.n0 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≠ ∅ ) |
5 |
|
disjf1.dj |
⊢ ( 𝜑 → Disj 𝑥 ∈ 𝐴 𝐵 ) |
6 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴 |
7 |
1 6
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) |
8 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 |
9 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑉 |
10 |
8 9
|
nfel |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ 𝑉 |
11 |
7 10
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) |
12 |
|
eleq1w |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) ) |
13 |
12
|
anbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) ) ) |
14 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑦 → 𝐵 = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
15 |
14
|
eleq1d |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 ∈ 𝑉 ↔ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) ) |
16 |
13 15
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) ) ) |
17 |
11 16 3
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) |
18 |
17
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) |
19 |
|
inidm |
⊢ ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 |
20 |
19
|
eqcomi |
⊢ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
21 |
20
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ ¬ 𝑦 = 𝑧 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) ) |
22 |
|
ineq2 |
⊢ ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 → ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
23 |
22
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ ¬ 𝑦 = 𝑧 ) → ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
24 |
|
nfcv |
⊢ Ⅎ 𝑤 𝐵 |
25 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑤 / 𝑥 ⦌ 𝐵 |
26 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑤 → 𝐵 = ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
27 |
24 25 26
|
cbvdisj |
⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑤 ∈ 𝐴 ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
28 |
5 27
|
sylib |
⊢ ( 𝜑 → Disj 𝑤 ∈ 𝐴 ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
29 |
28
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ ¬ 𝑦 = 𝑧 ) → Disj 𝑤 ∈ 𝐴 ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
30 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ ¬ 𝑦 = 𝑧 ) → ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) |
31 |
|
neqne |
⊢ ( ¬ 𝑦 = 𝑧 → 𝑦 ≠ 𝑧 ) |
32 |
31
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ ¬ 𝑦 = 𝑧 ) → 𝑦 ≠ 𝑧 ) |
33 |
|
csbeq1 |
⊢ ( 𝑤 = 𝑦 → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
34 |
|
csbeq1 |
⊢ ( 𝑤 = 𝑧 → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
35 |
33 34
|
disji2 |
⊢ ( ( Disj 𝑤 ∈ 𝐴 ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑦 ≠ 𝑧 ) → ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) = ∅ ) |
36 |
29 30 32 35
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ ¬ 𝑦 = 𝑧 ) → ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) = ∅ ) |
37 |
21 23 36
|
3eqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ ¬ 𝑦 = 𝑧 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ∅ ) |
38 |
|
nfcv |
⊢ Ⅎ 𝑥 ∅ |
39 |
8 38
|
nfne |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≠ ∅ |
40 |
7 39
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≠ ∅ ) |
41 |
14
|
neeq1d |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 ≠ ∅ ↔ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≠ ∅ ) ) |
42 |
13 41
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≠ ∅ ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≠ ∅ ) ) ) |
43 |
40 42 4
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≠ ∅ ) |
44 |
43
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≠ ∅ ) |
45 |
44
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ ¬ 𝑦 = 𝑧 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≠ ∅ ) |
46 |
45
|
neneqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ ¬ 𝑦 = 𝑧 ) → ¬ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ∅ ) |
47 |
37 46
|
condan |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) → 𝑦 = 𝑧 ) |
48 |
47
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 → 𝑦 = 𝑧 ) ) |
49 |
48
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 → 𝑦 = 𝑧 ) ) |
50 |
18 49
|
jca |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐴 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 → 𝑦 = 𝑧 ) ) ) |
51 |
|
nfcv |
⊢ Ⅎ 𝑦 𝐵 |
52 |
51 8 14
|
cbvmpt |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐴 ↦ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
53 |
2 52
|
eqtri |
⊢ 𝐹 = ( 𝑦 ∈ 𝐴 ↦ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
54 |
|
csbeq1 |
⊢ ( 𝑦 = 𝑧 → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
55 |
53 54
|
f1mpt |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝑉 ↔ ( ∀ 𝑦 ∈ 𝐴 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ⦋ 𝑦 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 → 𝑦 = 𝑧 ) ) ) |
56 |
50 55
|
sylibr |
⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1→ 𝑉 ) |