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 32731
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 9022 . 2 (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2 nfv 1914 . . . . . 6 𝑔(𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴)
3 nfv 1914 . . . . . 6 𝑔𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))
4 isfinite2 9334 . . . . . . . . . 10 (𝐴 ≺ ω → 𝐴 ∈ Fin)
5 isfinite4 14401 . . . . . . . . . 10 (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴)
64, 5sylib 218 . . . . . . . . 9 (𝐴 ≺ ω → (1...(♯‘𝐴)) ≈ 𝐴)
76adantr 480 . . . . . . . 8 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (1...(♯‘𝐴)) ≈ 𝐴)
8 bren 8995 . . . . . . . 8 ((1...(♯‘𝐴)) ≈ 𝐴 ↔ ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
97, 8sylib 218 . . . . . . 7 ((𝐴 ≺ ω ∧ 𝑍𝑉) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
1093adant3 1133 . . . . . 6 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
11 f1of 6848 . . . . . . . . . . . 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 5747 . . . . . . . . . . . . 13 ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)
1413eqcomi 2746 . . . . . . . . . . . 12 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
15 simplr 769 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑍𝑉)
16 fconst2g 7223 . . . . . . . . . . . . 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 6770 . . . . . . . . . . 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 838 . . . . . . . . . 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 13595 . . . . . . . . . . . 12 (1...(♯‘𝐴)) ⊆ ℕ
24 undif 4482 . . . . . . . . . . . 12 ((1...(♯‘𝐴)) ⊆ ℕ ↔ ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ)
2523, 24mpbi 230 . . . . . . . . . . 11 ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ
2625feq2i 6728 . . . . . . . . . 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 218 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
28273adantl3 1169 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
29 ssid 4006 . . . . . . . . . . . . 13 𝐴𝐴
30 simpr 484 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
31 f1ofo 6855 . . . . . . . . . . . . . 14 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴𝑔:(1...(♯‘𝐴))–onto𝐴)
32 forn 6823 . . . . . . . . . . . . . 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 4027 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑔)
3534orcd 874 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝐴 ⊆ ran 𝑔𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
36 ssun 4195 . . . . . . . . . . 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 6165 . . . . . . . . . 10 ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
3937, 38sseqtrrdi 4025 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
40393adantl3 1169 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
41 dff1o3 6854 . . . . . . . . . . 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 6162 . . . . . . . . . . . . 13 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
4544reseq1i 5993 . . . . . . . . . . . 12 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46 resundir 6012 . . . . . . . . . . . 12 ((𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
4745, 46eqtri 2765 . . . . . . . . . . 11 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
48 dff1o4 6856 . . . . . . . . . . . . . . . 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 6687 . . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ¬ 𝑍𝐴)
5414cnveqi 5885 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
55 cnvxp 6177 . . . . . . . . . . . . . . . . 17 ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
5654, 55eqtri 2765 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
5756reseq1i 5993 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴)
58 incom 4209 . . . . . . . . . . . . . . . . 17 (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59 disjsn 4711 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍𝐴)
6059biimpri 228 . . . . . . . . . . . . . . . . 17 𝑍𝐴 → (𝐴 ∩ {𝑍}) = ∅)
6158, 60eqtr3id 2791 . . . . . . . . . . . . . . . 16 𝑍𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62 xpdisjres 32611 . . . . . . . . . . . . . . . 16 (({𝑍} ∩ 𝐴) = ∅ → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
6361, 62syl 17 . . . . . . . . . . . . . . 15 𝑍𝐴 → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
6457, 63eqtrid 2789 . . . . . . . . . . . . . 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 4169 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = (𝑔 ∪ ∅))
67 un0 4394 . . . . . . . . . . . 12 (𝑔 ∪ ∅) = 𝑔
6866, 67eqtrdi 2793 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = 𝑔)
6947, 68eqtrid 2789 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = 𝑔)
7069funeqd 6588 . . . . . . . . 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 3484 . . . . . . . . . 10 𝑔 ∈ V
73 nnex 12272 . . . . . . . . . . . 12 ℕ ∈ V
74 difexg 5329 . . . . . . . . . . . 12 (ℕ ∈ V → (ℕ ∖ (1...(♯‘𝐴))) ∈ V)
7573, 74ax-mp 5 . . . . . . . . . . 11 (ℕ ∖ (1...(♯‘𝐴))) ∈ V
7675mptex 7243 . . . . . . . . . 10 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ∈ V
7772, 76unex 7764 . . . . . . . . 9 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ∈ V
78 feq1 6716 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79 rneq 5947 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
8079sseq2d 4016 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))))
81 cnveq 5884 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
82 eqidd 2738 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
8381, 82reseq12d 5998 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓𝐴) = ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
8483funeqd 6588 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (Fun (𝑓𝐴) ↔ Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
8578, 80, 843anbi123d 1438 . . . . . . . . 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 3606 . . . . . . . 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 1373 . . . . . . 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 2219 . . . . 5 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
90893expia 1122 . . . 4 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
91 nnenom 14021 . . . . . . . 8 ℕ ≈ ω
92 simpl 482 . . . . . . . . 9 ((𝐴 ≈ ω ∧ 𝑍𝑉) → 𝐴 ≈ ω)
9392ensymd 9045 . . . . . . . 8 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ω ≈ 𝐴)
94 entr 9046 . . . . . . . 8 ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
9591, 93, 94sylancr 587 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ℕ ≈ 𝐴)
96 bren 8995 . . . . . . 7 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
9795, 96sylib 218 . . . . . 6 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
98 nfv 1914 . . . . . . 7 𝑓(𝐴 ≈ ω ∧ 𝑍𝑉)
99 simpr 484 . . . . . . . . . 10 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–1-1-onto𝐴)
100 f1of 6848 . . . . . . . . . 10 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
101 ssun1 4178 . . . . . . . . . . 11 𝐴 ⊆ (𝐴 ∪ {𝑍})
102 fss 6752 . . . . . . . . . . 11 ((𝑓:ℕ⟶𝐴𝐴 ⊆ (𝐴 ∪ {𝑍})) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
103101, 102mpan2 691 . . . . . . . . . 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 6855 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
106 forn 6823 . . . . . . . . . . 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 4027 . . . . . . . . 9 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑓)
109 f1ocnv 6860 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:𝐴1-1-onto→ℕ)
110 f1of1 6847 . . . . . . . . . . 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 6862 . . . . . . . . . . 11 ((𝑓:𝐴1-1→ℕ ∧ 𝐴𝐴) → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
11329, 112mpan2 691 . . . . . . . . . 10 (𝑓:𝐴1-1→ℕ → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
114 f1ofun 6850 . . . . . . . . . 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 1129 . . . . . . . 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 2216 . . . . . 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 959 . . 3 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
1221213impia 1118 . 2 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
1231, 122syl3an1b 1405 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 206  wa 395  wo 848  w3a 1087   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626   class class class wbr 5143  cmpt 5225   × cxp 5683  ccnv 5684  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  1-1wf1 6558  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  ωcom 7887  cen 8982  cdom 8983  csdm 8984  Fincfn 8985  1c1 11156  cn 12266  ...cfz 13547  chash 14369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370
This theorem is referenced by:  carsggect  34320
  Copyright terms: Public domain W3C validator