Step |
Hyp |
Ref |
Expression |
1 |
|
fpwwe2.1 |
⊢ 𝑊 = { 〈 𝑥 , 𝑟 〉 ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) } |
2 |
|
fpwwe2.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
fpwwe2.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ∧ 𝑟 We 𝑥 ) ) → ( 𝑥 𝐹 𝑟 ) ∈ 𝐴 ) |
4 |
|
fpwwe2lem8.x |
⊢ ( 𝜑 → 𝑋 𝑊 𝑅 ) |
5 |
|
fpwwe2lem8.y |
⊢ ( 𝜑 → 𝑌 𝑊 𝑆 ) |
6 |
|
fpwwe2lem8.m |
⊢ 𝑀 = OrdIso ( 𝑅 , 𝑋 ) |
7 |
|
fpwwe2lem8.n |
⊢ 𝑁 = OrdIso ( 𝑆 , 𝑌 ) |
8 |
|
fpwwe2lem5.1 |
⊢ ( 𝜑 → 𝐵 ∈ dom 𝑀 ) |
9 |
|
fpwwe2lem5.2 |
⊢ ( 𝜑 → 𝐵 ∈ dom 𝑁 ) |
10 |
|
fpwwe2lem5.3 |
⊢ ( 𝜑 → ( 𝑀 ↾ 𝐵 ) = ( 𝑁 ↾ 𝐵 ) ) |
11 |
1 2
|
fpwwe2lem2 |
⊢ ( 𝜑 → ( 𝑋 𝑊 𝑅 ↔ ( ( 𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ ( 𝑋 × 𝑋 ) ) ∧ ( 𝑅 We 𝑋 ∧ ∀ 𝑦 ∈ 𝑋 [ ( ◡ 𝑅 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑅 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) ) ) |
12 |
4 11
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ ( 𝑋 × 𝑋 ) ) ∧ ( 𝑅 We 𝑋 ∧ ∀ 𝑦 ∈ 𝑋 [ ( ◡ 𝑅 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑅 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) ) |
13 |
12
|
simplrd |
⊢ ( 𝜑 → 𝑅 ⊆ ( 𝑋 × 𝑋 ) ) |
14 |
13
|
ssbrd |
⊢ ( 𝜑 → ( 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) → 𝐶 ( 𝑋 × 𝑋 ) ( 𝑀 ‘ 𝐵 ) ) ) |
15 |
|
brxp |
⊢ ( 𝐶 ( 𝑋 × 𝑋 ) ( 𝑀 ‘ 𝐵 ) ↔ ( 𝐶 ∈ 𝑋 ∧ ( 𝑀 ‘ 𝐵 ) ∈ 𝑋 ) ) |
16 |
15
|
simplbi |
⊢ ( 𝐶 ( 𝑋 × 𝑋 ) ( 𝑀 ‘ 𝐵 ) → 𝐶 ∈ 𝑋 ) |
17 |
14 16
|
syl6 |
⊢ ( 𝜑 → ( 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) → 𝐶 ∈ 𝑋 ) ) |
18 |
17
|
imp |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝐶 ∈ 𝑋 ) |
19 |
|
imassrn |
⊢ ( 𝑁 “ 𝐵 ) ⊆ ran 𝑁 |
20 |
1
|
relopabi |
⊢ Rel 𝑊 |
21 |
20
|
brrelex1i |
⊢ ( 𝑌 𝑊 𝑆 → 𝑌 ∈ V ) |
22 |
5 21
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ V ) |
23 |
1 2
|
fpwwe2lem2 |
⊢ ( 𝜑 → ( 𝑌 𝑊 𝑆 ↔ ( ( 𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ ( 𝑌 × 𝑌 ) ) ∧ ( 𝑆 We 𝑌 ∧ ∀ 𝑦 ∈ 𝑌 [ ( ◡ 𝑆 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑆 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) ) ) |
24 |
5 23
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ ( 𝑌 × 𝑌 ) ) ∧ ( 𝑆 We 𝑌 ∧ ∀ 𝑦 ∈ 𝑌 [ ( ◡ 𝑆 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑆 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) ) |
25 |
24
|
simprld |
⊢ ( 𝜑 → 𝑆 We 𝑌 ) |
26 |
7
|
oiiso |
⊢ ( ( 𝑌 ∈ V ∧ 𝑆 We 𝑌 ) → 𝑁 Isom E , 𝑆 ( dom 𝑁 , 𝑌 ) ) |
27 |
22 25 26
|
syl2anc |
⊢ ( 𝜑 → 𝑁 Isom E , 𝑆 ( dom 𝑁 , 𝑌 ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝑁 Isom E , 𝑆 ( dom 𝑁 , 𝑌 ) ) |
29 |
|
isof1o |
⊢ ( 𝑁 Isom E , 𝑆 ( dom 𝑁 , 𝑌 ) → 𝑁 : dom 𝑁 –1-1-onto→ 𝑌 ) |
30 |
28 29
|
syl |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝑁 : dom 𝑁 –1-1-onto→ 𝑌 ) |
31 |
|
f1ofo |
⊢ ( 𝑁 : dom 𝑁 –1-1-onto→ 𝑌 → 𝑁 : dom 𝑁 –onto→ 𝑌 ) |
32 |
|
forn |
⊢ ( 𝑁 : dom 𝑁 –onto→ 𝑌 → ran 𝑁 = 𝑌 ) |
33 |
30 31 32
|
3syl |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ran 𝑁 = 𝑌 ) |
34 |
19 33
|
sseqtrid |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( 𝑁 “ 𝐵 ) ⊆ 𝑌 ) |
35 |
20
|
brrelex1i |
⊢ ( 𝑋 𝑊 𝑅 → 𝑋 ∈ V ) |
36 |
4 35
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
37 |
12
|
simprld |
⊢ ( 𝜑 → 𝑅 We 𝑋 ) |
38 |
6
|
oiiso |
⊢ ( ( 𝑋 ∈ V ∧ 𝑅 We 𝑋 ) → 𝑀 Isom E , 𝑅 ( dom 𝑀 , 𝑋 ) ) |
39 |
36 37 38
|
syl2anc |
⊢ ( 𝜑 → 𝑀 Isom E , 𝑅 ( dom 𝑀 , 𝑋 ) ) |
40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝑀 Isom E , 𝑅 ( dom 𝑀 , 𝑋 ) ) |
41 |
|
isof1o |
⊢ ( 𝑀 Isom E , 𝑅 ( dom 𝑀 , 𝑋 ) → 𝑀 : dom 𝑀 –1-1-onto→ 𝑋 ) |
42 |
40 41
|
syl |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝑀 : dom 𝑀 –1-1-onto→ 𝑋 ) |
43 |
|
f1ocnvfv2 |
⊢ ( ( 𝑀 : dom 𝑀 –1-1-onto→ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝑀 ‘ ( ◡ 𝑀 ‘ 𝐶 ) ) = 𝐶 ) |
44 |
42 18 43
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( 𝑀 ‘ ( ◡ 𝑀 ‘ 𝐶 ) ) = 𝐶 ) |
45 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) |
46 |
44 45
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( 𝑀 ‘ ( ◡ 𝑀 ‘ 𝐶 ) ) 𝑅 ( 𝑀 ‘ 𝐵 ) ) |
47 |
|
f1ocnv |
⊢ ( 𝑀 : dom 𝑀 –1-1-onto→ 𝑋 → ◡ 𝑀 : 𝑋 –1-1-onto→ dom 𝑀 ) |
48 |
|
f1of |
⊢ ( ◡ 𝑀 : 𝑋 –1-1-onto→ dom 𝑀 → ◡ 𝑀 : 𝑋 ⟶ dom 𝑀 ) |
49 |
42 47 48
|
3syl |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ◡ 𝑀 : 𝑋 ⟶ dom 𝑀 ) |
50 |
49 18
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ◡ 𝑀 ‘ 𝐶 ) ∈ dom 𝑀 ) |
51 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝐵 ∈ dom 𝑀 ) |
52 |
|
isorel |
⊢ ( ( 𝑀 Isom E , 𝑅 ( dom 𝑀 , 𝑋 ) ∧ ( ( ◡ 𝑀 ‘ 𝐶 ) ∈ dom 𝑀 ∧ 𝐵 ∈ dom 𝑀 ) ) → ( ( ◡ 𝑀 ‘ 𝐶 ) E 𝐵 ↔ ( 𝑀 ‘ ( ◡ 𝑀 ‘ 𝐶 ) ) 𝑅 ( 𝑀 ‘ 𝐵 ) ) ) |
53 |
40 50 51 52
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ( ◡ 𝑀 ‘ 𝐶 ) E 𝐵 ↔ ( 𝑀 ‘ ( ◡ 𝑀 ‘ 𝐶 ) ) 𝑅 ( 𝑀 ‘ 𝐵 ) ) ) |
54 |
46 53
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ◡ 𝑀 ‘ 𝐶 ) E 𝐵 ) |
55 |
|
epelg |
⊢ ( 𝐵 ∈ dom 𝑀 → ( ( ◡ 𝑀 ‘ 𝐶 ) E 𝐵 ↔ ( ◡ 𝑀 ‘ 𝐶 ) ∈ 𝐵 ) ) |
56 |
51 55
|
syl |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ( ◡ 𝑀 ‘ 𝐶 ) E 𝐵 ↔ ( ◡ 𝑀 ‘ 𝐶 ) ∈ 𝐵 ) ) |
57 |
54 56
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ◡ 𝑀 ‘ 𝐶 ) ∈ 𝐵 ) |
58 |
|
ffn |
⊢ ( ◡ 𝑀 : 𝑋 ⟶ dom 𝑀 → ◡ 𝑀 Fn 𝑋 ) |
59 |
|
elpreima |
⊢ ( ◡ 𝑀 Fn 𝑋 → ( 𝐶 ∈ ( ◡ ◡ 𝑀 “ 𝐵 ) ↔ ( 𝐶 ∈ 𝑋 ∧ ( ◡ 𝑀 ‘ 𝐶 ) ∈ 𝐵 ) ) ) |
60 |
49 58 59
|
3syl |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( 𝐶 ∈ ( ◡ ◡ 𝑀 “ 𝐵 ) ↔ ( 𝐶 ∈ 𝑋 ∧ ( ◡ 𝑀 ‘ 𝐶 ) ∈ 𝐵 ) ) ) |
61 |
18 57 60
|
mpbir2and |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝐶 ∈ ( ◡ ◡ 𝑀 “ 𝐵 ) ) |
62 |
|
imacnvcnv |
⊢ ( ◡ ◡ 𝑀 “ 𝐵 ) = ( 𝑀 “ 𝐵 ) |
63 |
61 62
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝐶 ∈ ( 𝑀 “ 𝐵 ) ) |
64 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( 𝑀 ↾ 𝐵 ) = ( 𝑁 ↾ 𝐵 ) ) |
65 |
64
|
rneqd |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ran ( 𝑀 ↾ 𝐵 ) = ran ( 𝑁 ↾ 𝐵 ) ) |
66 |
|
df-ima |
⊢ ( 𝑀 “ 𝐵 ) = ran ( 𝑀 ↾ 𝐵 ) |
67 |
|
df-ima |
⊢ ( 𝑁 “ 𝐵 ) = ran ( 𝑁 ↾ 𝐵 ) |
68 |
65 66 67
|
3eqtr4g |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( 𝑀 “ 𝐵 ) = ( 𝑁 “ 𝐵 ) ) |
69 |
63 68
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝐶 ∈ ( 𝑁 “ 𝐵 ) ) |
70 |
34 69
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → 𝐶 ∈ 𝑌 ) |
71 |
64
|
cnveqd |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ◡ ( 𝑀 ↾ 𝐵 ) = ◡ ( 𝑁 ↾ 𝐵 ) ) |
72 |
|
dff1o3 |
⊢ ( 𝑀 : dom 𝑀 –1-1-onto→ 𝑋 ↔ ( 𝑀 : dom 𝑀 –onto→ 𝑋 ∧ Fun ◡ 𝑀 ) ) |
73 |
72
|
simprbi |
⊢ ( 𝑀 : dom 𝑀 –1-1-onto→ 𝑋 → Fun ◡ 𝑀 ) |
74 |
|
funcnvres |
⊢ ( Fun ◡ 𝑀 → ◡ ( 𝑀 ↾ 𝐵 ) = ( ◡ 𝑀 ↾ ( 𝑀 “ 𝐵 ) ) ) |
75 |
42 73 74
|
3syl |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ◡ ( 𝑀 ↾ 𝐵 ) = ( ◡ 𝑀 ↾ ( 𝑀 “ 𝐵 ) ) ) |
76 |
|
dff1o3 |
⊢ ( 𝑁 : dom 𝑁 –1-1-onto→ 𝑌 ↔ ( 𝑁 : dom 𝑁 –onto→ 𝑌 ∧ Fun ◡ 𝑁 ) ) |
77 |
76
|
simprbi |
⊢ ( 𝑁 : dom 𝑁 –1-1-onto→ 𝑌 → Fun ◡ 𝑁 ) |
78 |
|
funcnvres |
⊢ ( Fun ◡ 𝑁 → ◡ ( 𝑁 ↾ 𝐵 ) = ( ◡ 𝑁 ↾ ( 𝑁 “ 𝐵 ) ) ) |
79 |
30 77 78
|
3syl |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ◡ ( 𝑁 ↾ 𝐵 ) = ( ◡ 𝑁 ↾ ( 𝑁 “ 𝐵 ) ) ) |
80 |
71 75 79
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ◡ 𝑀 ↾ ( 𝑀 “ 𝐵 ) ) = ( ◡ 𝑁 ↾ ( 𝑁 “ 𝐵 ) ) ) |
81 |
80
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ( ◡ 𝑀 ↾ ( 𝑀 “ 𝐵 ) ) ‘ 𝐶 ) = ( ( ◡ 𝑁 ↾ ( 𝑁 “ 𝐵 ) ) ‘ 𝐶 ) ) |
82 |
63
|
fvresd |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ( ◡ 𝑀 ↾ ( 𝑀 “ 𝐵 ) ) ‘ 𝐶 ) = ( ◡ 𝑀 ‘ 𝐶 ) ) |
83 |
69
|
fvresd |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ( ◡ 𝑁 ↾ ( 𝑁 “ 𝐵 ) ) ‘ 𝐶 ) = ( ◡ 𝑁 ‘ 𝐶 ) ) |
84 |
81 82 83
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( ◡ 𝑀 ‘ 𝐶 ) = ( ◡ 𝑁 ‘ 𝐶 ) ) |
85 |
18 70 84
|
3jca |
⊢ ( ( 𝜑 ∧ 𝐶 𝑅 ( 𝑀 ‘ 𝐵 ) ) → ( 𝐶 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ ( ◡ 𝑀 ‘ 𝐶 ) = ( ◡ 𝑁 ‘ 𝐶 ) ) ) |