Step |
Hyp |
Ref |
Expression |
1 |
|
frege124d.f |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
2 |
|
frege124d.x |
⊢ ( 𝜑 → 𝑋 ∈ dom 𝐹 ) |
3 |
|
frege124d.a |
⊢ ( 𝜑 → 𝐴 = ( 𝐹 ‘ 𝑋 ) ) |
4 |
|
frege124d.xb |
⊢ ( 𝜑 → 𝑋 ( t+ ‘ 𝐹 ) 𝐵 ) |
5 |
|
frege124d.fun |
⊢ ( 𝜑 → Fun 𝐹 ) |
6 |
3
|
eqcomd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) = 𝐴 ) |
7 |
|
funbrfvb |
⊢ ( ( Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝑋 ) = 𝐴 ↔ 𝑋 𝐹 𝐴 ) ) |
8 |
5 2 7
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑋 ) = 𝐴 ↔ 𝑋 𝐹 𝐴 ) ) |
9 |
6 8
|
mpbid |
⊢ ( 𝜑 → 𝑋 𝐹 𝐴 ) |
10 |
|
funeu |
⊢ ( ( Fun 𝐹 ∧ 𝑋 𝐹 𝐴 ) → ∃! 𝑎 𝑋 𝐹 𝑎 ) |
11 |
5 9 10
|
syl2anc |
⊢ ( 𝜑 → ∃! 𝑎 𝑋 𝐹 𝑎 ) |
12 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑋 ) ∈ V |
13 |
3 12
|
eqeltrdi |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
14 |
|
sbcan |
⊢ ( [ 𝐴 / 𝑎 ] ( 𝑋 𝐹 𝑎 ∧ ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ↔ ( [ 𝐴 / 𝑎 ] 𝑋 𝐹 𝑎 ∧ [ 𝐴 / 𝑎 ] ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
15 |
|
sbcbr2g |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] 𝑋 𝐹 𝑎 ↔ 𝑋 𝐹 ⦋ 𝐴 / 𝑎 ⦌ 𝑎 ) ) |
16 |
|
csbvarg |
⊢ ( 𝐴 ∈ V → ⦋ 𝐴 / 𝑎 ⦌ 𝑎 = 𝐴 ) |
17 |
16
|
breq2d |
⊢ ( 𝐴 ∈ V → ( 𝑋 𝐹 ⦋ 𝐴 / 𝑎 ⦌ 𝑎 ↔ 𝑋 𝐹 𝐴 ) ) |
18 |
15 17
|
bitrd |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] 𝑋 𝐹 𝑎 ↔ 𝑋 𝐹 𝐴 ) ) |
19 |
|
sbcng |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ↔ ¬ [ 𝐴 / 𝑎 ] 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
20 |
|
sbcbr1g |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ↔ ⦋ 𝐴 / 𝑎 ⦌ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
21 |
16
|
breq1d |
⊢ ( 𝐴 ∈ V → ( ⦋ 𝐴 / 𝑎 ⦌ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ↔ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
22 |
20 21
|
bitrd |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ↔ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
23 |
22
|
notbid |
⊢ ( 𝐴 ∈ V → ( ¬ [ 𝐴 / 𝑎 ] 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ↔ ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
24 |
19 23
|
bitrd |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ↔ ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
25 |
18 24
|
anbi12d |
⊢ ( 𝐴 ∈ V → ( ( [ 𝐴 / 𝑎 ] 𝑋 𝐹 𝑎 ∧ [ 𝐴 / 𝑎 ] ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ↔ ( 𝑋 𝐹 𝐴 ∧ ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
26 |
14 25
|
syl5bb |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑎 ] ( 𝑋 𝐹 𝑎 ∧ ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ↔ ( 𝑋 𝐹 𝐴 ∧ ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
27 |
13 26
|
syl |
⊢ ( 𝜑 → ( [ 𝐴 / 𝑎 ] ( 𝑋 𝐹 𝑎 ∧ ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ↔ ( 𝑋 𝐹 𝐴 ∧ ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
28 |
|
spesbc |
⊢ ( [ 𝐴 / 𝑎 ] ( 𝑋 𝐹 𝑎 ∧ ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) → ∃ 𝑎 ( 𝑋 𝐹 𝑎 ∧ ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
29 |
27 28
|
syl6bir |
⊢ ( 𝜑 → ( ( 𝑋 𝐹 𝐴 ∧ ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) → ∃ 𝑎 ( 𝑋 𝐹 𝑎 ∧ ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
30 |
9 29
|
mpand |
⊢ ( 𝜑 → ( ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 → ∃ 𝑎 ( 𝑋 𝐹 𝑎 ∧ ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
31 |
|
eupicka |
⊢ ( ( ∃! 𝑎 𝑋 𝐹 𝑎 ∧ ∃ 𝑎 ( 𝑋 𝐹 𝑎 ∧ ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) → ∀ 𝑎 ( 𝑋 𝐹 𝑎 → ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
32 |
11 30 31
|
syl6an |
⊢ ( 𝜑 → ( ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 → ∀ 𝑎 ( 𝑋 𝐹 𝑎 → ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
33 |
|
alinexa |
⊢ ( ∀ 𝑎 ( 𝑋 𝐹 𝑎 → ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ↔ ¬ ∃ 𝑎 ( 𝑋 𝐹 𝑎 ∧ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
34 |
|
funrel |
⊢ ( Fun 𝐹 → Rel 𝐹 ) |
35 |
5 34
|
syl |
⊢ ( 𝜑 → Rel 𝐹 ) |
36 |
|
reltrclfv |
⊢ ( ( 𝐹 ∈ V ∧ Rel 𝐹 ) → Rel ( t+ ‘ 𝐹 ) ) |
37 |
1 35 36
|
syl2anc |
⊢ ( 𝜑 → Rel ( t+ ‘ 𝐹 ) ) |
38 |
|
brrelex2 |
⊢ ( ( Rel ( t+ ‘ 𝐹 ) ∧ 𝑋 ( t+ ‘ 𝐹 ) 𝐵 ) → 𝐵 ∈ V ) |
39 |
37 4 38
|
syl2anc |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
40 |
|
brcog |
⊢ ( ( 𝑋 ∈ dom 𝐹 ∧ 𝐵 ∈ V ) → ( 𝑋 ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) 𝐵 ↔ ∃ 𝑎 ( 𝑋 𝐹 𝑎 ∧ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
41 |
2 39 40
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) 𝐵 ↔ ∃ 𝑎 ( 𝑋 𝐹 𝑎 ∧ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
42 |
41
|
notbid |
⊢ ( 𝜑 → ( ¬ 𝑋 ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) 𝐵 ↔ ¬ ∃ 𝑎 ( 𝑋 𝐹 𝑎 ∧ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
43 |
33 42
|
bitr4id |
⊢ ( 𝜑 → ( ∀ 𝑎 ( 𝑋 𝐹 𝑎 → ¬ 𝑎 ( t+ ‘ 𝐹 ) 𝐵 ) ↔ ¬ 𝑋 ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) 𝐵 ) ) |
44 |
32 43
|
sylibd |
⊢ ( 𝜑 → ( ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 → ¬ 𝑋 ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) 𝐵 ) ) |
45 |
|
brdif |
⊢ ( 𝑋 ( ( t+ ‘ 𝐹 ) ∖ ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ) 𝐵 ↔ ( 𝑋 ( t+ ‘ 𝐹 ) 𝐵 ∧ ¬ 𝑋 ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) 𝐵 ) ) |
46 |
45
|
simplbi2 |
⊢ ( 𝑋 ( t+ ‘ 𝐹 ) 𝐵 → ( ¬ 𝑋 ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) 𝐵 → 𝑋 ( ( t+ ‘ 𝐹 ) ∖ ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ) 𝐵 ) ) |
47 |
4 44 46
|
sylsyld |
⊢ ( 𝜑 → ( ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 → 𝑋 ( ( t+ ‘ 𝐹 ) ∖ ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ) 𝐵 ) ) |
48 |
|
trclfvdecomr |
⊢ ( 𝐹 ∈ V → ( t+ ‘ 𝐹 ) = ( 𝐹 ∪ ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ) ) |
49 |
1 48
|
syl |
⊢ ( 𝜑 → ( t+ ‘ 𝐹 ) = ( 𝐹 ∪ ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ) ) |
50 |
|
uncom |
⊢ ( 𝐹 ∪ ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ) = ( ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ∪ 𝐹 ) |
51 |
49 50
|
eqtrdi |
⊢ ( 𝜑 → ( t+ ‘ 𝐹 ) = ( ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ∪ 𝐹 ) ) |
52 |
|
eqimss |
⊢ ( ( t+ ‘ 𝐹 ) = ( ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ∪ 𝐹 ) → ( t+ ‘ 𝐹 ) ⊆ ( ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ∪ 𝐹 ) ) |
53 |
51 52
|
syl |
⊢ ( 𝜑 → ( t+ ‘ 𝐹 ) ⊆ ( ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ∪ 𝐹 ) ) |
54 |
|
ssundif |
⊢ ( ( t+ ‘ 𝐹 ) ⊆ ( ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ∪ 𝐹 ) ↔ ( ( t+ ‘ 𝐹 ) ∖ ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ) ⊆ 𝐹 ) |
55 |
53 54
|
sylib |
⊢ ( 𝜑 → ( ( t+ ‘ 𝐹 ) ∖ ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ) ⊆ 𝐹 ) |
56 |
55
|
ssbrd |
⊢ ( 𝜑 → ( 𝑋 ( ( t+ ‘ 𝐹 ) ∖ ( ( t+ ‘ 𝐹 ) ∘ 𝐹 ) ) 𝐵 → 𝑋 𝐹 𝐵 ) ) |
57 |
47 56
|
syld |
⊢ ( 𝜑 → ( ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 → 𝑋 𝐹 𝐵 ) ) |
58 |
|
funbrfv |
⊢ ( Fun 𝐹 → ( 𝑋 𝐹 𝐵 → ( 𝐹 ‘ 𝑋 ) = 𝐵 ) ) |
59 |
5 57 58
|
sylsyld |
⊢ ( 𝜑 → ( ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 → ( 𝐹 ‘ 𝑋 ) = 𝐵 ) ) |
60 |
|
eqcom |
⊢ ( ( 𝐹 ‘ 𝑋 ) = 𝐵 ↔ 𝐵 = ( 𝐹 ‘ 𝑋 ) ) |
61 |
59 60
|
syl6ib |
⊢ ( 𝜑 → ( ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 → 𝐵 = ( 𝐹 ‘ 𝑋 ) ) ) |
62 |
|
eqtr3 |
⊢ ( ( 𝐴 = ( 𝐹 ‘ 𝑋 ) ∧ 𝐵 = ( 𝐹 ‘ 𝑋 ) ) → 𝐴 = 𝐵 ) |
63 |
3 61 62
|
syl6an |
⊢ ( 𝜑 → ( ¬ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 → 𝐴 = 𝐵 ) ) |
64 |
63
|
orrd |
⊢ ( 𝜑 → ( 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐴 = 𝐵 ) ) |