Step |
Hyp |
Ref |
Expression |
1 |
|
2ndresdju.u |
⊢ 𝑈 = ∪ 𝑥 ∈ 𝑋 ( { 𝑥 } × 𝐶 ) |
2 |
|
2ndresdju.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
2ndresdju.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑊 ) |
4 |
|
2ndresdju.1 |
⊢ ( 𝜑 → Disj 𝑥 ∈ 𝑋 𝐶 ) |
5 |
|
2ndresdju.2 |
⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 ) |
6 |
|
fo2nd |
⊢ 2nd : V –onto→ V |
7 |
|
fofn |
⊢ ( 2nd : V –onto→ V → 2nd Fn V ) |
8 |
6 7
|
mp1i |
⊢ ( 𝜑 → 2nd Fn V ) |
9 |
|
ssv |
⊢ 𝑈 ⊆ V |
10 |
9
|
a1i |
⊢ ( 𝜑 → 𝑈 ⊆ V ) |
11 |
8 10
|
fnssresd |
⊢ ( 𝜑 → ( 2nd ↾ 𝑈 ) Fn 𝑈 ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → 𝑢 ∈ 𝑈 ) |
13 |
12
|
fvresd |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( 2nd ‘ 𝑢 ) ) |
14 |
|
djussxp2 |
⊢ ∪ 𝑥 ∈ 𝑋 ( { 𝑥 } × 𝐶 ) ⊆ ( 𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶 ) |
15 |
5
|
xpeq2d |
⊢ ( 𝜑 → ( 𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶 ) = ( 𝑋 × 𝐴 ) ) |
16 |
14 15
|
sseqtrid |
⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝑋 ( { 𝑥 } × 𝐶 ) ⊆ ( 𝑋 × 𝐴 ) ) |
17 |
1 16
|
eqsstrid |
⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑋 × 𝐴 ) ) |
18 |
17
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → 𝑢 ∈ ( 𝑋 × 𝐴 ) ) |
19 |
|
xp2nd |
⊢ ( 𝑢 ∈ ( 𝑋 × 𝐴 ) → ( 2nd ‘ 𝑢 ) ∈ 𝐴 ) |
20 |
18 19
|
syl |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( 2nd ‘ 𝑢 ) ∈ 𝐴 ) |
21 |
13 20
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) ∈ 𝐴 ) |
22 |
21
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑢 ∈ 𝑈 ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) ∈ 𝐴 ) |
23 |
|
ffnfv |
⊢ ( ( 2nd ↾ 𝑈 ) : 𝑈 ⟶ 𝐴 ↔ ( ( 2nd ↾ 𝑈 ) Fn 𝑈 ∧ ∀ 𝑢 ∈ 𝑈 ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) ∈ 𝐴 ) ) |
24 |
11 22 23
|
sylanbrc |
⊢ ( 𝜑 → ( 2nd ↾ 𝑈 ) : 𝑈 ⟶ 𝐴 ) |
25 |
|
nfv |
⊢ Ⅎ 𝑥 𝜑 |
26 |
|
nfiu1 |
⊢ Ⅎ 𝑥 ∪ 𝑥 ∈ 𝑋 ( { 𝑥 } × 𝐶 ) |
27 |
1 26
|
nfcxfr |
⊢ Ⅎ 𝑥 𝑈 |
28 |
27
|
nfcri |
⊢ Ⅎ 𝑥 𝑢 ∈ 𝑈 |
29 |
25 28
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) |
30 |
27
|
nfcri |
⊢ Ⅎ 𝑥 𝑣 ∈ 𝑈 |
31 |
29 30
|
nfan |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) |
32 |
|
nfcv |
⊢ Ⅎ 𝑥 2nd |
33 |
32 27
|
nfres |
⊢ Ⅎ 𝑥 ( 2nd ↾ 𝑈 ) |
34 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑢 |
35 |
33 34
|
nffv |
⊢ Ⅎ 𝑥 ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) |
36 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑣 |
37 |
33 36
|
nffv |
⊢ Ⅎ 𝑥 ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) |
38 |
35 37
|
nfeq |
⊢ Ⅎ 𝑥 ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) |
39 |
31 38
|
nfan |
⊢ Ⅎ 𝑥 ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) |
40 |
|
nfv |
⊢ Ⅎ 𝑥 𝑢 = 𝑣 |
41 |
1
|
eleq2i |
⊢ ( 𝑢 ∈ 𝑈 ↔ 𝑢 ∈ ∪ 𝑥 ∈ 𝑋 ( { 𝑥 } × 𝐶 ) ) |
42 |
|
eliunxp |
⊢ ( 𝑢 ∈ ∪ 𝑥 ∈ 𝑋 ( { 𝑥 } × 𝐶 ) ↔ ∃ 𝑥 ∃ 𝑐 ( 𝑢 = 〈 𝑥 , 𝑐 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑐 ∈ 𝐶 ) ) ) |
43 |
41 42
|
sylbb |
⊢ ( 𝑢 ∈ 𝑈 → ∃ 𝑥 ∃ 𝑐 ( 𝑢 = 〈 𝑥 , 𝑐 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑐 ∈ 𝐶 ) ) ) |
44 |
43
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) → ∃ 𝑥 ∃ 𝑐 ( 𝑢 = 〈 𝑥 , 𝑐 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑐 ∈ 𝐶 ) ) ) |
45 |
1
|
eleq2i |
⊢ ( 𝑣 ∈ 𝑈 ↔ 𝑣 ∈ ∪ 𝑥 ∈ 𝑋 ( { 𝑥 } × 𝐶 ) ) |
46 |
|
eliunxp |
⊢ ( 𝑣 ∈ ∪ 𝑥 ∈ 𝑋 ( { 𝑥 } × 𝐶 ) ↔ ∃ 𝑥 ∃ 𝑑 ( 𝑣 = 〈 𝑥 , 𝑑 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ 𝐶 ) ) ) |
47 |
45 46
|
bitri |
⊢ ( 𝑣 ∈ 𝑈 ↔ ∃ 𝑥 ∃ 𝑑 ( 𝑣 = 〈 𝑥 , 𝑑 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ 𝐶 ) ) ) |
48 |
|
nfv |
⊢ Ⅎ 𝑦 ∃ 𝑑 ( 𝑣 = 〈 𝑥 , 𝑑 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ 𝐶 ) ) |
49 |
|
nfv |
⊢ Ⅎ 𝑥 𝑣 = 〈 𝑦 , 𝑑 〉 |
50 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 ∈ 𝑋 |
51 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐶 |
52 |
51
|
nfcri |
⊢ Ⅎ 𝑥 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 |
53 |
50 52
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) |
54 |
49 53
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑣 = 〈 𝑦 , 𝑑 〉 ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) |
55 |
54
|
nfex |
⊢ Ⅎ 𝑥 ∃ 𝑑 ( 𝑣 = 〈 𝑦 , 𝑑 〉 ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) |
56 |
|
opeq1 |
⊢ ( 𝑥 = 𝑦 → 〈 𝑥 , 𝑑 〉 = 〈 𝑦 , 𝑑 〉 ) |
57 |
56
|
eqeq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝑣 = 〈 𝑥 , 𝑑 〉 ↔ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ) |
58 |
|
eleq1w |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑋 ↔ 𝑦 ∈ 𝑋 ) ) |
59 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑦 → 𝐶 = ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) |
60 |
59
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝑑 ∈ 𝐶 ↔ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) |
61 |
58 60
|
anbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ 𝐶 ) ↔ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) ) |
62 |
57 61
|
anbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑣 = 〈 𝑥 , 𝑑 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ 𝐶 ) ) ↔ ( 𝑣 = 〈 𝑦 , 𝑑 〉 ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) ) ) |
63 |
62
|
exbidv |
⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑑 ( 𝑣 = 〈 𝑥 , 𝑑 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ 𝐶 ) ) ↔ ∃ 𝑑 ( 𝑣 = 〈 𝑦 , 𝑑 〉 ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) ) ) |
64 |
48 55 63
|
cbvexv1 |
⊢ ( ∃ 𝑥 ∃ 𝑑 ( 𝑣 = 〈 𝑥 , 𝑑 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ 𝐶 ) ) ↔ ∃ 𝑦 ∃ 𝑑 ( 𝑣 = 〈 𝑦 , 𝑑 〉 ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) ) |
65 |
47 64
|
sylbb |
⊢ ( 𝑣 ∈ 𝑈 → ∃ 𝑦 ∃ 𝑑 ( 𝑣 = 〈 𝑦 , 𝑑 〉 ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) ) |
66 |
65
|
ad5antlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) → ∃ 𝑦 ∃ 𝑑 ( 𝑣 = 〈 𝑦 , 𝑑 〉 ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) ) |
67 |
4
|
ad9antr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → Disj 𝑥 ∈ 𝑋 𝐶 ) |
68 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑥 ∈ 𝑋 ) |
69 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑦 ∈ 𝑋 ) |
70 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑐 ∈ 𝐶 ) |
71 |
|
simp-7r |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) |
72 |
|
simp-9r |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑢 ∈ 𝑈 ) |
73 |
72
|
fvresd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( 2nd ‘ 𝑢 ) ) |
74 |
|
simp-6r |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑢 = 〈 𝑥 , 𝑐 〉 ) |
75 |
74
|
fveq2d |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 〈 𝑥 , 𝑐 〉 ) ) |
76 |
|
vex |
⊢ 𝑥 ∈ V |
77 |
|
vex |
⊢ 𝑐 ∈ V |
78 |
76 77
|
op2nd |
⊢ ( 2nd ‘ 〈 𝑥 , 𝑐 〉 ) = 𝑐 |
79 |
75 78
|
eqtrdi |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → ( 2nd ‘ 𝑢 ) = 𝑐 ) |
80 |
73 79
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = 𝑐 ) |
81 |
|
simp-8r |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑣 ∈ 𝑈 ) |
82 |
81
|
fvresd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) = ( 2nd ‘ 𝑣 ) ) |
83 |
|
simpllr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑣 = 〈 𝑦 , 𝑑 〉 ) |
84 |
83
|
fveq2d |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → ( 2nd ‘ 𝑣 ) = ( 2nd ‘ 〈 𝑦 , 𝑑 〉 ) ) |
85 |
|
vex |
⊢ 𝑦 ∈ V |
86 |
|
vex |
⊢ 𝑑 ∈ V |
87 |
85 86
|
op2nd |
⊢ ( 2nd ‘ 〈 𝑦 , 𝑑 〉 ) = 𝑑 |
88 |
84 87
|
eqtrdi |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → ( 2nd ‘ 𝑣 ) = 𝑑 ) |
89 |
82 88
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) = 𝑑 ) |
90 |
71 80 89
|
3eqtr3d |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑐 = 𝑑 ) |
91 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) |
92 |
90 91
|
eqeltrd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑐 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) |
93 |
51 59
|
disjif |
⊢ ( ( Disj 𝑥 ∈ 𝑋 𝐶 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) → 𝑥 = 𝑦 ) |
94 |
67 68 69 70 92 93
|
syl122anc |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑥 = 𝑦 ) |
95 |
94 90
|
opeq12d |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 〈 𝑥 , 𝑐 〉 = 〈 𝑦 , 𝑑 〉 ) |
96 |
95 74 83
|
3eqtr4d |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) → 𝑢 = 𝑣 ) |
97 |
96
|
anasss |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) ∧ 𝑣 = 〈 𝑦 , 𝑑 〉 ) ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) → 𝑢 = 𝑣 ) |
98 |
97
|
expl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) → ( ( 𝑣 = 〈 𝑦 , 𝑑 〉 ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) → 𝑢 = 𝑣 ) ) |
99 |
98
|
exlimdvv |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) → ( ∃ 𝑦 ∃ 𝑑 ( 𝑣 = 〈 𝑦 , 𝑑 〉 ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑑 ∈ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) → 𝑢 = 𝑣 ) ) |
100 |
66 99
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝐶 ) → 𝑢 = 𝑣 ) |
101 |
100
|
anasss |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) ∧ 𝑢 = 〈 𝑥 , 𝑐 〉 ) ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑐 ∈ 𝐶 ) ) → 𝑢 = 𝑣 ) |
102 |
101
|
expl |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) → ( ( 𝑢 = 〈 𝑥 , 𝑐 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑐 ∈ 𝐶 ) ) → 𝑢 = 𝑣 ) ) |
103 |
102
|
exlimdv |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) → ( ∃ 𝑐 ( 𝑢 = 〈 𝑥 , 𝑐 〉 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑐 ∈ 𝐶 ) ) → 𝑢 = 𝑣 ) ) |
104 |
39 40 44 103
|
exlimimdd |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) ) → 𝑢 = 𝑣 ) |
105 |
104
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) |
106 |
105
|
anasss |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) ) → ( ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) |
107 |
106
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑢 ∈ 𝑈 ∀ 𝑣 ∈ 𝑈 ( ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) |
108 |
|
dff13 |
⊢ ( ( 2nd ↾ 𝑈 ) : 𝑈 –1-1→ 𝐴 ↔ ( ( 2nd ↾ 𝑈 ) : 𝑈 ⟶ 𝐴 ∧ ∀ 𝑢 ∈ 𝑈 ∀ 𝑣 ∈ 𝑈 ( ( ( 2nd ↾ 𝑈 ) ‘ 𝑢 ) = ( ( 2nd ↾ 𝑈 ) ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) ) |
109 |
24 107 108
|
sylanbrc |
⊢ ( 𝜑 → ( 2nd ↾ 𝑈 ) : 𝑈 –1-1→ 𝐴 ) |