Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  padct Structured version   Visualization version   GIF version

Theorem padct 32514
Description: Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.)
Assertion
Ref Expression
padct ((𝐴 ≼ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
Distinct variable groups:   𝐴,𝑓   𝑓,𝑉   𝑓,𝑍

Proof of Theorem padct
Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brdom2 9003 . 2 (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2 nfv 1910 . . . . . 6 𝑔(𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴)
3 nfv 1910 . . . . . 6 𝑔𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))
4 isfinite2 9326 . . . . . . . . . 10 (𝐴 ≺ ω → 𝐴 ∈ Fin)
5 isfinite4 14354 . . . . . . . . . 10 (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴)
64, 5sylib 217 . . . . . . . . 9 (𝐴 ≺ ω → (1...(♯‘𝐴)) ≈ 𝐴)
76adantr 480 . . . . . . . 8 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (1...(♯‘𝐴)) ≈ 𝐴)
8 bren 8974 . . . . . . . 8 ((1...(♯‘𝐴)) ≈ 𝐴 ↔ ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
97, 8sylib 217 . . . . . . 7 ((𝐴 ≺ ω ∧ 𝑍𝑉) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
1093adant3 1130 . . . . . 6 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
11 f1of 6839 . . . . . . . . . . . 12 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴𝑔:(1...(♯‘𝐴))⟶𝐴)
1211adantl 481 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑔:(1...(♯‘𝐴))⟶𝐴)
13 fconstmpt 5740 . . . . . . . . . . . . 13 ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)
1413eqcomi 2737 . . . . . . . . . . . 12 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
15 simplr 768 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑍𝑉)
16 fconst2g 7215 . . . . . . . . . . . . 13 (𝑍𝑉 → ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(♯‘𝐴)))⟶{𝑍} ↔ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})))
1715, 16syl 17 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(♯‘𝐴)))⟶{𝑍} ↔ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})))
1814, 17mpbiri 258 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(♯‘𝐴)))⟶{𝑍})
19 disjdif 4472 . . . . . . . . . . . 12 ((1...(♯‘𝐴)) ∩ (ℕ ∖ (1...(♯‘𝐴)))) = ∅
2019a1i 11 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((1...(♯‘𝐴)) ∩ (ℕ ∖ (1...(♯‘𝐴)))) = ∅)
21 fun 6759 . . . . . . . . . . 11 (((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}))
2212, 18, 20, 21syl21anc 837 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴))))⟶(𝐴 ∪ {𝑍}))
23 fz1ssnn 13565 . . . . . . . . . . . 12 (1...(♯‘𝐴)) ⊆ ℕ
24 undif 4482 . . . . . . . . . . . 12 ((1...(♯‘𝐴)) ⊆ ℕ ↔ ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ)
2523, 24mpbi 229 . . . . . . . . . . 11 ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ
2625feq2i 6714 . . . . . . . . . 10 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴))))⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
2722, 26sylib 217 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
28273adantl3 1166 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
29 ssid 4002 . . . . . . . . . . . . 13 𝐴𝐴
30 simpr 484 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
31 f1ofo 6846 . . . . . . . . . . . . . 14 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴𝑔:(1...(♯‘𝐴))–onto𝐴)
32 forn 6814 . . . . . . . . . . . . . 14 (𝑔:(1...(♯‘𝐴))–onto𝐴 → ran 𝑔 = 𝐴)
3330, 31, 323syl 18 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ran 𝑔 = 𝐴)
3429, 33sseqtrrid 4033 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑔)
3534orcd 872 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝐴 ⊆ ran 𝑔𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
36 ssun 4189 . . . . . . . . . . 11 ((𝐴 ⊆ ran 𝑔𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝐴 ⊆ (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
3735, 36syl 17 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
38 rnun 6150 . . . . . . . . . 10 ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
3937, 38sseqtrrdi 4031 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
40393adantl3 1166 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
41 dff1o3 6845 . . . . . . . . . . 11 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴 ↔ (𝑔:(1...(♯‘𝐴))–onto𝐴 ∧ Fun 𝑔))
4241simprbi 496 . . . . . . . . . 10 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴 → Fun 𝑔)
4342adantl 481 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → Fun 𝑔)
44 cnvun 6147 . . . . . . . . . . . . 13 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
4544reseq1i 5981 . . . . . . . . . . . 12 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46 resundir 6000 . . . . . . . . . . . 12 ((𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
4745, 46eqtri 2756 . . . . . . . . . . 11 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
48 dff1o4 6847 . . . . . . . . . . . . . . . 16 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴 ↔ (𝑔 Fn (1...(♯‘𝐴)) ∧ 𝑔 Fn 𝐴))
4948simprbi 496 . . . . . . . . . . . . . . 15 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴𝑔 Fn 𝐴)
50 fnresdm 6674 . . . . . . . . . . . . . . 15 (𝑔 Fn 𝐴 → (𝑔𝐴) = 𝑔)
5149, 50syl 17 . . . . . . . . . . . . . 14 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝑔𝐴) = 𝑔)
5251adantl 481 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑔𝐴) = 𝑔)
53 simpl3 1191 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ¬ 𝑍𝐴)
5414cnveqi 5877 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
55 cnvxp 6161 . . . . . . . . . . . . . . . . 17 ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
5654, 55eqtri 2756 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
5756reseq1i 5981 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴)
58 incom 4201 . . . . . . . . . . . . . . . . 17 (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59 disjsn 4716 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍𝐴)
6059biimpri 227 . . . . . . . . . . . . . . . . 17 𝑍𝐴 → (𝐴 ∩ {𝑍}) = ∅)
6158, 60eqtr3id 2782 . . . . . . . . . . . . . . . 16 𝑍𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62 xpdisjres 32401 . . . . . . . . . . . . . . . 16 (({𝑍} ∩ 𝐴) = ∅ → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
6361, 62syl 17 . . . . . . . . . . . . . . 15 𝑍𝐴 → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
6457, 63eqtrid 2780 . . . . . . . . . . . . . 14 𝑍𝐴 → ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = ∅)
6553, 64syl 17 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = ∅)
6652, 65uneq12d 4163 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = (𝑔 ∪ ∅))
67 un0 4391 . . . . . . . . . . . 12 (𝑔 ∪ ∅) = 𝑔
6866, 67eqtrdi 2784 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = 𝑔)
6947, 68eqtrid 2780 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = 𝑔)
7069funeqd 6575 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) ↔ Fun 𝑔))
7143, 70mpbird 257 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
72 vex 3475 . . . . . . . . . 10 𝑔 ∈ V
73 nnex 12249 . . . . . . . . . . . 12 ℕ ∈ V
74 difexg 5329 . . . . . . . . . . . 12 (ℕ ∈ V → (ℕ ∖ (1...(♯‘𝐴))) ∈ V)
7573, 74ax-mp 5 . . . . . . . . . . 11 (ℕ ∖ (1...(♯‘𝐴))) ∈ V
7675mptex 7235 . . . . . . . . . 10 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ∈ V
7772, 76unex 7748 . . . . . . . . 9 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ∈ V
78 feq1 6703 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79 rneq 5938 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
8079sseq2d 4012 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))))
81 cnveq 5876 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
82 eqidd 2729 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
8381, 82reseq12d 5986 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓𝐴) = ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
8483funeqd 6575 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (Fun (𝑓𝐴) ↔ Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
8578, 80, 843anbi123d 1433 . . . . . . . . 9 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ((𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)) ↔ ((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))))
8677, 85spcev 3593 . . . . . . . 8 (((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 (𝑓𝐴)))
8728, 40, 71, 86syl3anc 1369 . . . . . . 7 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
8887ex 412 . . . . . 6 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
892, 3, 10, 88exlimimdd 2208 . . . . 5 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
90893expia 1119 . . . 4 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
91 nnenom 13978 . . . . . . . 8 ℕ ≈ ω
92 simpl 482 . . . . . . . . 9 ((𝐴 ≈ ω ∧ 𝑍𝑉) → 𝐴 ≈ ω)
9392ensymd 9026 . . . . . . . 8 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ω ≈ 𝐴)
94 entr 9027 . . . . . . . 8 ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
9591, 93, 94sylancr 586 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ℕ ≈ 𝐴)
96 bren 8974 . . . . . . 7 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
9795, 96sylib 217 . . . . . 6 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
98 nfv 1910 . . . . . . 7 𝑓(𝐴 ≈ ω ∧ 𝑍𝑉)
99 simpr 484 . . . . . . . . . 10 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–1-1-onto𝐴)
100 f1of 6839 . . . . . . . . . 10 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
101 ssun1 4172 . . . . . . . . . . 11 𝐴 ⊆ (𝐴 ∪ {𝑍})
102 fss 6739 . . . . . . . . . . 11 ((𝑓:ℕ⟶𝐴𝐴 ⊆ (𝐴 ∪ {𝑍})) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
103101, 102mpan2 690 . . . . . . . . . 10 (𝑓:ℕ⟶𝐴𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
10499, 100, 1033syl 18 . . . . . . . . 9 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
105 f1ofo 6846 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
106 forn 6814 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → ran 𝑓 = 𝐴)
10799, 105, 1063syl 18 . . . . . . . . . 10 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → ran 𝑓 = 𝐴)
10829, 107sseqtrrid 4033 . . . . . . . . 9 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑓)
109 f1ocnv 6851 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:𝐴1-1-onto→ℕ)
110 f1of1 6838 . . . . . . . . . . 11 (𝑓:𝐴1-1-onto→ℕ → 𝑓:𝐴1-1→ℕ)
11199, 109, 1103syl 18 . . . . . . . . . 10 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:𝐴1-1→ℕ)
112 f1ores 6853 . . . . . . . . . . 11 ((𝑓:𝐴1-1→ℕ ∧ 𝐴𝐴) → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
11329, 112mpan2 690 . . . . . . . . . 10 (𝑓:𝐴1-1→ℕ → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
114 f1ofun 6841 . . . . . . . . . 10 ((𝑓𝐴):𝐴1-1-onto→(𝑓𝐴) → Fun (𝑓𝐴))
115111, 113, 1143syl 18 . . . . . . . . 9 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → Fun (𝑓𝐴))
116104, 108, 1153jca 1126 . . . . . . . 8 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
117116ex 412 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → (𝑓:ℕ–1-1-onto𝐴 → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
11898, 117eximd 2205 . . . . . 6 ((𝐴 ≈ ω ∧ 𝑍𝑉) → (∃𝑓 𝑓:ℕ–1-1-onto𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
11997, 118mpd 15 . . . . 5 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
120119a1d 25 . . . 4 ((𝐴 ≈ ω ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
12190, 120jaoian 955 . . 3 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
1221213impia 1115 . 2 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
1231, 122syl3an1b 1401 1 ((𝐴 ≼ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846  w3a 1085   = wceq 1534  wex 1774  wcel 2099  Vcvv 3471  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4323  {csn 4629   class class class wbr 5148  cmpt 5231   × cxp 5676  ccnv 5677  ran crn 5679  cres 5680  cima 5681  Fun wfun 6542   Fn wfn 6543  wf 6544  1-1wf1 6545  ontowfo 6546  1-1-ontowf1o 6547  cfv 6548  (class class class)co 7420  ωcom 7870  cen 8961  cdom 8962  csdm 8963  Fincfn 8964  1c1 11140  cn 12243  ...cfz 13517  chash 14322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-n0 12504  df-z 12590  df-uz 12854  df-fz 13518  df-hash 14323
This theorem is referenced by:  carsggect  33938
  Copyright terms: Public domain W3C validator