Step |
Hyp |
Ref |
Expression |
1 |
|
nnfoctb |
⊢ ( ( 𝐴 ≼ ω ∧ 𝐴 ≠ ∅ ) → ∃ 𝑔 𝑔 : ℕ –onto→ 𝐴 ) |
2 |
|
fofn |
⊢ ( 𝑔 : ℕ –onto→ 𝐴 → 𝑔 Fn ℕ ) |
3 |
|
nnex |
⊢ ℕ ∈ V |
4 |
3
|
a1i |
⊢ ( 𝑔 : ℕ –onto→ 𝐴 → ℕ ∈ V ) |
5 |
|
ltwenn |
⊢ < We ℕ |
6 |
5
|
a1i |
⊢ ( 𝑔 : ℕ –onto→ 𝐴 → < We ℕ ) |
7 |
2 4 6
|
wessf1orn |
⊢ ( 𝑔 : ℕ –onto→ 𝐴 → ∃ 𝑥 ∈ 𝒫 ℕ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) |
8 |
|
f1odm |
⊢ ( ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 → dom ( 𝑔 ↾ 𝑥 ) = 𝑥 ) |
9 |
8
|
adantl |
⊢ ( ( 𝑥 ∈ 𝒫 ℕ ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → dom ( 𝑔 ↾ 𝑥 ) = 𝑥 ) |
10 |
|
elpwi |
⊢ ( 𝑥 ∈ 𝒫 ℕ → 𝑥 ⊆ ℕ ) |
11 |
10
|
adantr |
⊢ ( ( 𝑥 ∈ 𝒫 ℕ ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → 𝑥 ⊆ ℕ ) |
12 |
9 11
|
eqsstrd |
⊢ ( ( 𝑥 ∈ 𝒫 ℕ ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → dom ( 𝑔 ↾ 𝑥 ) ⊆ ℕ ) |
13 |
12
|
3adant1 |
⊢ ( ( 𝑔 : ℕ –onto→ 𝐴 ∧ 𝑥 ∈ 𝒫 ℕ ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → dom ( 𝑔 ↾ 𝑥 ) ⊆ ℕ ) |
14 |
|
simpr |
⊢ ( ( 𝑔 : ℕ –onto→ 𝐴 ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) |
15 |
|
eqidd |
⊢ ( ( 𝑔 : ℕ –onto→ 𝐴 ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → ( 𝑔 ↾ 𝑥 ) = ( 𝑔 ↾ 𝑥 ) ) |
16 |
8
|
eqcomd |
⊢ ( ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 → 𝑥 = dom ( 𝑔 ↾ 𝑥 ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝑔 : ℕ –onto→ 𝐴 ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → 𝑥 = dom ( 𝑔 ↾ 𝑥 ) ) |
18 |
|
forn |
⊢ ( 𝑔 : ℕ –onto→ 𝐴 → ran 𝑔 = 𝐴 ) |
19 |
18
|
adantr |
⊢ ( ( 𝑔 : ℕ –onto→ 𝐴 ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → ran 𝑔 = 𝐴 ) |
20 |
15 17 19
|
f1oeq123d |
⊢ ( ( 𝑔 : ℕ –onto→ 𝐴 ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → ( ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ↔ ( 𝑔 ↾ 𝑥 ) : dom ( 𝑔 ↾ 𝑥 ) –1-1-onto→ 𝐴 ) ) |
21 |
14 20
|
mpbid |
⊢ ( ( 𝑔 : ℕ –onto→ 𝐴 ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → ( 𝑔 ↾ 𝑥 ) : dom ( 𝑔 ↾ 𝑥 ) –1-1-onto→ 𝐴 ) |
22 |
21
|
3adant2 |
⊢ ( ( 𝑔 : ℕ –onto→ 𝐴 ∧ 𝑥 ∈ 𝒫 ℕ ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → ( 𝑔 ↾ 𝑥 ) : dom ( 𝑔 ↾ 𝑥 ) –1-1-onto→ 𝐴 ) |
23 |
|
vex |
⊢ 𝑔 ∈ V |
24 |
23
|
resex |
⊢ ( 𝑔 ↾ 𝑥 ) ∈ V |
25 |
|
dmeq |
⊢ ( 𝑓 = ( 𝑔 ↾ 𝑥 ) → dom 𝑓 = dom ( 𝑔 ↾ 𝑥 ) ) |
26 |
25
|
sseq1d |
⊢ ( 𝑓 = ( 𝑔 ↾ 𝑥 ) → ( dom 𝑓 ⊆ ℕ ↔ dom ( 𝑔 ↾ 𝑥 ) ⊆ ℕ ) ) |
27 |
|
id |
⊢ ( 𝑓 = ( 𝑔 ↾ 𝑥 ) → 𝑓 = ( 𝑔 ↾ 𝑥 ) ) |
28 |
|
eqidd |
⊢ ( 𝑓 = ( 𝑔 ↾ 𝑥 ) → 𝐴 = 𝐴 ) |
29 |
27 25 28
|
f1oeq123d |
⊢ ( 𝑓 = ( 𝑔 ↾ 𝑥 ) → ( 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ↔ ( 𝑔 ↾ 𝑥 ) : dom ( 𝑔 ↾ 𝑥 ) –1-1-onto→ 𝐴 ) ) |
30 |
26 29
|
anbi12d |
⊢ ( 𝑓 = ( 𝑔 ↾ 𝑥 ) → ( ( dom 𝑓 ⊆ ℕ ∧ 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ) ↔ ( dom ( 𝑔 ↾ 𝑥 ) ⊆ ℕ ∧ ( 𝑔 ↾ 𝑥 ) : dom ( 𝑔 ↾ 𝑥 ) –1-1-onto→ 𝐴 ) ) ) |
31 |
24 30
|
spcev |
⊢ ( ( dom ( 𝑔 ↾ 𝑥 ) ⊆ ℕ ∧ ( 𝑔 ↾ 𝑥 ) : dom ( 𝑔 ↾ 𝑥 ) –1-1-onto→ 𝐴 ) → ∃ 𝑓 ( dom 𝑓 ⊆ ℕ ∧ 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ) ) |
32 |
13 22 31
|
syl2anc |
⊢ ( ( 𝑔 : ℕ –onto→ 𝐴 ∧ 𝑥 ∈ 𝒫 ℕ ∧ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 ) → ∃ 𝑓 ( dom 𝑓 ⊆ ℕ ∧ 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ) ) |
33 |
32
|
3exp |
⊢ ( 𝑔 : ℕ –onto→ 𝐴 → ( 𝑥 ∈ 𝒫 ℕ → ( ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 → ∃ 𝑓 ( dom 𝑓 ⊆ ℕ ∧ 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ) ) ) ) |
34 |
33
|
rexlimdv |
⊢ ( 𝑔 : ℕ –onto→ 𝐴 → ( ∃ 𝑥 ∈ 𝒫 ℕ ( 𝑔 ↾ 𝑥 ) : 𝑥 –1-1-onto→ ran 𝑔 → ∃ 𝑓 ( dom 𝑓 ⊆ ℕ ∧ 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ) ) ) |
35 |
7 34
|
mpd |
⊢ ( 𝑔 : ℕ –onto→ 𝐴 → ∃ 𝑓 ( dom 𝑓 ⊆ ℕ ∧ 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ) ) |
36 |
35
|
a1i |
⊢ ( ( 𝐴 ≼ ω ∧ 𝐴 ≠ ∅ ) → ( 𝑔 : ℕ –onto→ 𝐴 → ∃ 𝑓 ( dom 𝑓 ⊆ ℕ ∧ 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ) ) ) |
37 |
36
|
exlimdv |
⊢ ( ( 𝐴 ≼ ω ∧ 𝐴 ≠ ∅ ) → ( ∃ 𝑔 𝑔 : ℕ –onto→ 𝐴 → ∃ 𝑓 ( dom 𝑓 ⊆ ℕ ∧ 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ) ) ) |
38 |
1 37
|
mpd |
⊢ ( ( 𝐴 ≼ ω ∧ 𝐴 ≠ ∅ ) → ∃ 𝑓 ( dom 𝑓 ⊆ ℕ ∧ 𝑓 : dom 𝑓 –1-1-onto→ 𝐴 ) ) |