| Step |
Hyp |
Ref |
Expression |
| 1 |
|
brdom2 |
⊢ ( 𝐴 ≼ ω ↔ ( 𝐴 ≺ ω ∨ 𝐴 ≈ ω ) ) |
| 2 |
|
nfv |
⊢ Ⅎ 𝑔 ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) |
| 3 |
|
nfv |
⊢ Ⅎ 𝑔 ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) |
| 4 |
|
isfinite2 |
⊢ ( 𝐴 ≺ ω → 𝐴 ∈ Fin ) |
| 5 |
|
isfinite4 |
⊢ ( 𝐴 ∈ Fin ↔ ( 1 ... ( ♯ ‘ 𝐴 ) ) ≈ 𝐴 ) |
| 6 |
4 5
|
sylib |
⊢ ( 𝐴 ≺ ω → ( 1 ... ( ♯ ‘ 𝐴 ) ) ≈ 𝐴 ) |
| 7 |
6
|
adantr |
⊢ ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) → ( 1 ... ( ♯ ‘ 𝐴 ) ) ≈ 𝐴 ) |
| 8 |
|
bren |
⊢ ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ≈ 𝐴 ↔ ∃ 𝑔 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) |
| 9 |
7 8
|
sylib |
⊢ ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) → ∃ 𝑔 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) |
| 10 |
9
|
3adant3 |
⊢ ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) → ∃ 𝑔 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) |
| 11 |
|
f1of |
⊢ ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
| 12 |
11
|
adantl |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
| 13 |
|
fconstmpt |
⊢ ( ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) × { 𝑍 } ) = ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) |
| 14 |
13
|
eqcomi |
⊢ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) = ( ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) × { 𝑍 } ) |
| 15 |
|
simplr |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → 𝑍 ∈ 𝑉 ) |
| 16 |
|
fconst2g |
⊢ ( 𝑍 ∈ 𝑉 → ( ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) : ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ⟶ { 𝑍 } ↔ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) = ( ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) × { 𝑍 } ) ) ) |
| 17 |
15 16
|
syl |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) : ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ⟶ { 𝑍 } ↔ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) = ( ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) × { 𝑍 } ) ) ) |
| 18 |
14 17
|
mpbiri |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) : ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ⟶ { 𝑍 } ) |
| 19 |
|
disjdif |
⊢ ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ∩ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) = ∅ |
| 20 |
19
|
a1i |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ∩ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) = ∅ ) |
| 21 |
|
fun |
⊢ ( ( ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) : ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ⟶ { 𝑍 } ) ∧ ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ∩ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) = ∅ ) → ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) : ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ∪ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) ⟶ ( 𝐴 ∪ { 𝑍 } ) ) |
| 22 |
12 18 20 21
|
syl21anc |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) : ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ∪ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) ⟶ ( 𝐴 ∪ { 𝑍 } ) ) |
| 23 |
|
fz1ssnn |
⊢ ( 1 ... ( ♯ ‘ 𝐴 ) ) ⊆ ℕ |
| 24 |
|
undif |
⊢ ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ⊆ ℕ ↔ ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ∪ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) = ℕ ) |
| 25 |
23 24
|
mpbi |
⊢ ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ∪ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) = ℕ |
| 26 |
25
|
feq2i |
⊢ ( ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) : ( ( 1 ... ( ♯ ‘ 𝐴 ) ) ∪ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) ⟶ ( 𝐴 ∪ { 𝑍 } ) ↔ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ) |
| 27 |
22 26
|
sylib |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ) |
| 28 |
27
|
3adantl3 |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ) |
| 29 |
|
ssid |
⊢ 𝐴 ⊆ 𝐴 |
| 30 |
|
simpr |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) |
| 31 |
|
f1ofo |
⊢ ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –onto→ 𝐴 ) |
| 32 |
|
forn |
⊢ ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –onto→ 𝐴 → ran 𝑔 = 𝐴 ) |
| 33 |
30 31 32
|
3syl |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ran 𝑔 = 𝐴 ) |
| 34 |
29 33
|
sseqtrrid |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → 𝐴 ⊆ ran 𝑔 ) |
| 35 |
34
|
orcd |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( 𝐴 ⊆ ran 𝑔 ∨ 𝐴 ⊆ ran ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ) |
| 36 |
|
ssun |
⊢ ( ( 𝐴 ⊆ ran 𝑔 ∨ 𝐴 ⊆ ran ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) → 𝐴 ⊆ ( ran 𝑔 ∪ ran ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ) |
| 37 |
35 36
|
syl |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → 𝐴 ⊆ ( ran 𝑔 ∪ ran ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ) |
| 38 |
|
rnun |
⊢ ran ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) = ( ran 𝑔 ∪ ran ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) |
| 39 |
37 38
|
sseqtrrdi |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → 𝐴 ⊆ ran ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ) |
| 40 |
39
|
3adantl3 |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → 𝐴 ⊆ ran ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ) |
| 41 |
|
dff1o3 |
⊢ ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ↔ ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –onto→ 𝐴 ∧ Fun ◡ 𝑔 ) ) |
| 42 |
41
|
simprbi |
⊢ ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → Fun ◡ 𝑔 ) |
| 43 |
42
|
adantl |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → Fun ◡ 𝑔 ) |
| 44 |
|
cnvun |
⊢ ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) = ( ◡ 𝑔 ∪ ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) |
| 45 |
44
|
reseq1i |
⊢ ( ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) = ( ( ◡ 𝑔 ∪ ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) |
| 46 |
|
resundir |
⊢ ( ( ◡ 𝑔 ∪ ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) = ( ( ◡ 𝑔 ↾ 𝐴 ) ∪ ( ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ↾ 𝐴 ) ) |
| 47 |
45 46
|
eqtri |
⊢ ( ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) = ( ( ◡ 𝑔 ↾ 𝐴 ) ∪ ( ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ↾ 𝐴 ) ) |
| 48 |
|
dff1o4 |
⊢ ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ↔ ( 𝑔 Fn ( 1 ... ( ♯ ‘ 𝐴 ) ) ∧ ◡ 𝑔 Fn 𝐴 ) ) |
| 49 |
48
|
simprbi |
⊢ ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ◡ 𝑔 Fn 𝐴 ) |
| 50 |
|
fnresdm |
⊢ ( ◡ 𝑔 Fn 𝐴 → ( ◡ 𝑔 ↾ 𝐴 ) = ◡ 𝑔 ) |
| 51 |
49 50
|
syl |
⊢ ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ( ◡ 𝑔 ↾ 𝐴 ) = ◡ 𝑔 ) |
| 52 |
51
|
adantl |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( ◡ 𝑔 ↾ 𝐴 ) = ◡ 𝑔 ) |
| 53 |
|
simpl3 |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ¬ 𝑍 ∈ 𝐴 ) |
| 54 |
14
|
cnveqi |
⊢ ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) = ◡ ( ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) × { 𝑍 } ) |
| 55 |
|
cnvxp |
⊢ ◡ ( ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) × { 𝑍 } ) = ( { 𝑍 } × ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) |
| 56 |
54 55
|
eqtri |
⊢ ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) = ( { 𝑍 } × ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) |
| 57 |
56
|
reseq1i |
⊢ ( ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ↾ 𝐴 ) = ( ( { 𝑍 } × ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) ↾ 𝐴 ) |
| 58 |
|
incom |
⊢ ( 𝐴 ∩ { 𝑍 } ) = ( { 𝑍 } ∩ 𝐴 ) |
| 59 |
|
disjsn |
⊢ ( ( 𝐴 ∩ { 𝑍 } ) = ∅ ↔ ¬ 𝑍 ∈ 𝐴 ) |
| 60 |
59
|
biimpri |
⊢ ( ¬ 𝑍 ∈ 𝐴 → ( 𝐴 ∩ { 𝑍 } ) = ∅ ) |
| 61 |
58 60
|
eqtr3id |
⊢ ( ¬ 𝑍 ∈ 𝐴 → ( { 𝑍 } ∩ 𝐴 ) = ∅ ) |
| 62 |
|
xpdisjres |
⊢ ( ( { 𝑍 } ∩ 𝐴 ) = ∅ → ( ( { 𝑍 } × ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) ↾ 𝐴 ) = ∅ ) |
| 63 |
61 62
|
syl |
⊢ ( ¬ 𝑍 ∈ 𝐴 → ( ( { 𝑍 } × ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ) ↾ 𝐴 ) = ∅ ) |
| 64 |
57 63
|
eqtrid |
⊢ ( ¬ 𝑍 ∈ 𝐴 → ( ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ↾ 𝐴 ) = ∅ ) |
| 65 |
53 64
|
syl |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ↾ 𝐴 ) = ∅ ) |
| 66 |
52 65
|
uneq12d |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( ( ◡ 𝑔 ↾ 𝐴 ) ∪ ( ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ↾ 𝐴 ) ) = ( ◡ 𝑔 ∪ ∅ ) ) |
| 67 |
|
un0 |
⊢ ( ◡ 𝑔 ∪ ∅ ) = ◡ 𝑔 |
| 68 |
66 67
|
eqtrdi |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( ( ◡ 𝑔 ↾ 𝐴 ) ∪ ( ◡ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ↾ 𝐴 ) ) = ◡ 𝑔 ) |
| 69 |
47 68
|
eqtrid |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) = ◡ 𝑔 ) |
| 70 |
69
|
funeqd |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( Fun ( ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) ↔ Fun ◡ 𝑔 ) ) |
| 71 |
43 70
|
mpbird |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → Fun ( ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) ) |
| 72 |
|
vex |
⊢ 𝑔 ∈ V |
| 73 |
|
nnex |
⊢ ℕ ∈ V |
| 74 |
|
difexg |
⊢ ( ℕ ∈ V → ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ∈ V ) |
| 75 |
73 74
|
ax-mp |
⊢ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ∈ V |
| 76 |
75
|
mptex |
⊢ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ∈ V |
| 77 |
72 76
|
unex |
⊢ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ∈ V |
| 78 |
|
feq1 |
⊢ ( 𝑓 = ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) → ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ↔ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ) ) |
| 79 |
|
rneq |
⊢ ( 𝑓 = ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) → ran 𝑓 = ran ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ) |
| 80 |
79
|
sseq2d |
⊢ ( 𝑓 = ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) → ( 𝐴 ⊆ ran 𝑓 ↔ 𝐴 ⊆ ran ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ) ) |
| 81 |
|
cnveq |
⊢ ( 𝑓 = ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) → ◡ 𝑓 = ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ) |
| 82 |
|
eqidd |
⊢ ( 𝑓 = ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) → 𝐴 = 𝐴 ) |
| 83 |
81 82
|
reseq12d |
⊢ ( 𝑓 = ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) → ( ◡ 𝑓 ↾ 𝐴 ) = ( ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) ) |
| 84 |
83
|
funeqd |
⊢ ( 𝑓 = ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) → ( Fun ( ◡ 𝑓 ↾ 𝐴 ) ↔ Fun ( ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) ) ) |
| 85 |
78 80 84
|
3anbi123d |
⊢ ( 𝑓 = ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) → ( ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ↔ ( ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ∧ Fun ( ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) ) ) ) |
| 86 |
77 85
|
spcev |
⊢ ( ( ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ∧ Fun ( ◡ ( 𝑔 ∪ ( 𝑥 ∈ ( ℕ ∖ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) ↦ 𝑍 ) ) ↾ 𝐴 ) ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) |
| 87 |
28 40 71 86
|
syl3anc |
⊢ ( ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) ∧ 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) |
| 88 |
87
|
ex |
⊢ ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) → ( 𝑔 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) ) |
| 89 |
2 3 10 88
|
exlimimdd |
⊢ ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) |
| 90 |
89
|
3expia |
⊢ ( ( 𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ) → ( ¬ 𝑍 ∈ 𝐴 → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) ) |
| 91 |
|
nnenom |
⊢ ℕ ≈ ω |
| 92 |
|
simpl |
⊢ ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) → 𝐴 ≈ ω ) |
| 93 |
92
|
ensymd |
⊢ ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) → ω ≈ 𝐴 ) |
| 94 |
|
entr |
⊢ ( ( ℕ ≈ ω ∧ ω ≈ 𝐴 ) → ℕ ≈ 𝐴 ) |
| 95 |
91 93 94
|
sylancr |
⊢ ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) → ℕ ≈ 𝐴 ) |
| 96 |
|
bren |
⊢ ( ℕ ≈ 𝐴 ↔ ∃ 𝑓 𝑓 : ℕ –1-1-onto→ 𝐴 ) |
| 97 |
95 96
|
sylib |
⊢ ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) → ∃ 𝑓 𝑓 : ℕ –1-1-onto→ 𝐴 ) |
| 98 |
|
nfv |
⊢ Ⅎ 𝑓 ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) |
| 99 |
|
simpr |
⊢ ( ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑓 : ℕ –1-1-onto→ 𝐴 ) → 𝑓 : ℕ –1-1-onto→ 𝐴 ) |
| 100 |
|
f1of |
⊢ ( 𝑓 : ℕ –1-1-onto→ 𝐴 → 𝑓 : ℕ ⟶ 𝐴 ) |
| 101 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ { 𝑍 } ) |
| 102 |
|
fss |
⊢ ( ( 𝑓 : ℕ ⟶ 𝐴 ∧ 𝐴 ⊆ ( 𝐴 ∪ { 𝑍 } ) ) → 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ) |
| 103 |
101 102
|
mpan2 |
⊢ ( 𝑓 : ℕ ⟶ 𝐴 → 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ) |
| 104 |
99 100 103
|
3syl |
⊢ ( ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑓 : ℕ –1-1-onto→ 𝐴 ) → 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ) |
| 105 |
|
f1ofo |
⊢ ( 𝑓 : ℕ –1-1-onto→ 𝐴 → 𝑓 : ℕ –onto→ 𝐴 ) |
| 106 |
|
forn |
⊢ ( 𝑓 : ℕ –onto→ 𝐴 → ran 𝑓 = 𝐴 ) |
| 107 |
99 105 106
|
3syl |
⊢ ( ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑓 : ℕ –1-1-onto→ 𝐴 ) → ran 𝑓 = 𝐴 ) |
| 108 |
29 107
|
sseqtrrid |
⊢ ( ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑓 : ℕ –1-1-onto→ 𝐴 ) → 𝐴 ⊆ ran 𝑓 ) |
| 109 |
|
f1ocnv |
⊢ ( 𝑓 : ℕ –1-1-onto→ 𝐴 → ◡ 𝑓 : 𝐴 –1-1-onto→ ℕ ) |
| 110 |
|
f1of1 |
⊢ ( ◡ 𝑓 : 𝐴 –1-1-onto→ ℕ → ◡ 𝑓 : 𝐴 –1-1→ ℕ ) |
| 111 |
99 109 110
|
3syl |
⊢ ( ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑓 : ℕ –1-1-onto→ 𝐴 ) → ◡ 𝑓 : 𝐴 –1-1→ ℕ ) |
| 112 |
|
f1ores |
⊢ ( ( ◡ 𝑓 : 𝐴 –1-1→ ℕ ∧ 𝐴 ⊆ 𝐴 ) → ( ◡ 𝑓 ↾ 𝐴 ) : 𝐴 –1-1-onto→ ( ◡ 𝑓 “ 𝐴 ) ) |
| 113 |
29 112
|
mpan2 |
⊢ ( ◡ 𝑓 : 𝐴 –1-1→ ℕ → ( ◡ 𝑓 ↾ 𝐴 ) : 𝐴 –1-1-onto→ ( ◡ 𝑓 “ 𝐴 ) ) |
| 114 |
|
f1ofun |
⊢ ( ( ◡ 𝑓 ↾ 𝐴 ) : 𝐴 –1-1-onto→ ( ◡ 𝑓 “ 𝐴 ) → Fun ( ◡ 𝑓 ↾ 𝐴 ) ) |
| 115 |
111 113 114
|
3syl |
⊢ ( ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑓 : ℕ –1-1-onto→ 𝐴 ) → Fun ( ◡ 𝑓 ↾ 𝐴 ) ) |
| 116 |
104 108 115
|
3jca |
⊢ ( ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑓 : ℕ –1-1-onto→ 𝐴 ) → ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) |
| 117 |
116
|
ex |
⊢ ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) → ( 𝑓 : ℕ –1-1-onto→ 𝐴 → ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) ) |
| 118 |
98 117
|
eximd |
⊢ ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) → ( ∃ 𝑓 𝑓 : ℕ –1-1-onto→ 𝐴 → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) ) |
| 119 |
97 118
|
mpd |
⊢ ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) |
| 120 |
119
|
a1d |
⊢ ( ( 𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉 ) → ( ¬ 𝑍 ∈ 𝐴 → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) ) |
| 121 |
90 120
|
jaoian |
⊢ ( ( ( 𝐴 ≺ ω ∨ 𝐴 ≈ ω ) ∧ 𝑍 ∈ 𝑉 ) → ( ¬ 𝑍 ∈ 𝐴 → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) ) |
| 122 |
121
|
3impia |
⊢ ( ( ( 𝐴 ≺ ω ∨ 𝐴 ≈ ω ) ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) |
| 123 |
1 122
|
syl3an1b |
⊢ ( ( 𝐴 ≼ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴 ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ ( 𝐴 ∪ { 𝑍 } ) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun ( ◡ 𝑓 ↾ 𝐴 ) ) ) |