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 29625
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 8027 . 2 (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2 isfinite2 8259 . . . . . . . . . 10 (𝐴 ≺ ω → 𝐴 ∈ Fin)
3 isfinite4 13191 . . . . . . . . . 10 (𝐴 ∈ Fin ↔ (1...(#‘𝐴)) ≈ 𝐴)
42, 3sylib 208 . . . . . . . . 9 (𝐴 ≺ ω → (1...(#‘𝐴)) ≈ 𝐴)
54adantr 480 . . . . . . . 8 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (1...(#‘𝐴)) ≈ 𝐴)
6 bren 8006 . . . . . . . 8 ((1...(#‘𝐴)) ≈ 𝐴 ↔ ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto𝐴)
75, 6sylib 208 . . . . . . 7 ((𝐴 ≺ ω ∧ 𝑍𝑉) → ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto𝐴)
873adant3 1101 . . . . . 6 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto𝐴)
9 nfv 1883 . . . . . . 7 𝑔(𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴)
10 nfv 1883 . . . . . . 7 𝑔𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))
11 f1of 6175 . . . . . . . . . . . . 13 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴𝑔:(1...(#‘𝐴))⟶𝐴)
1211adantl 481 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝑔:(1...(#‘𝐴))⟶𝐴)
13 fconstmpt 5197 . . . . . . . . . . . . . 14 ((ℕ ∖ (1...(#‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)
1413eqcomi 2660 . . . . . . . . . . . . 13 (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(#‘𝐴))) × {𝑍})
15 simplr 807 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝑍𝑉)
16 fconst2g 6509 . . . . . . . . . . . . . 14 (𝑍𝑉 → ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(#‘𝐴)))⟶{𝑍} ↔ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(#‘𝐴))) × {𝑍})))
1715, 16syl 17 . . . . . . . . . . . . 13 (((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 248 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(#‘𝐴)))⟶{𝑍})
19 disjdif 4073 . . . . . . . . . . . . 13 ((1...(#‘𝐴)) ∩ (ℕ ∖ (1...(#‘𝐴)))) = ∅
2019a1i 11 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ((1...(#‘𝐴)) ∩ (ℕ ∖ (1...(#‘𝐴)))) = ∅)
21 fun 6104 . . . . . . . . . . . 12 (((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 1365 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴))))⟶(𝐴 ∪ {𝑍}))
23 fz1ssnn 12410 . . . . . . . . . . . . 13 (1...(#‘𝐴)) ⊆ ℕ
24 undif 4082 . . . . . . . . . . . . 13 ((1...(#‘𝐴)) ⊆ ℕ ↔ ((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴)))) = ℕ)
2523, 24mpbi 220 . . . . . . . . . . . 12 ((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴)))) = ℕ
2625feq2i 6075 . . . . . . . . . . 11 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴))))⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
2722, 26sylib 208 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
28273adantl3 1239 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
29 ssid 3657 . . . . . . . . . . . . . 14 𝐴𝐴
30 simpr 476 . . . . . . . . . . . . . . 15 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝑔:(1...(#‘𝐴))–1-1-onto𝐴)
31 f1ofo 6182 . . . . . . . . . . . . . . 15 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴𝑔:(1...(#‘𝐴))–onto𝐴)
32 forn 6156 . . . . . . . . . . . . . . 15 (𝑔:(1...(#‘𝐴))–onto𝐴 → ran 𝑔 = 𝐴)
3330, 31, 323syl 18 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ran 𝑔 = 𝐴)
3429, 33syl5sseqr 3687 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑔)
3534orcd 406 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (𝐴 ⊆ ran 𝑔𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
36 ssun 3825 . . . . . . . . . . . 12 ((𝐴 ⊆ ran 𝑔𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → 𝐴 ⊆ (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
3735, 36syl 17 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
38 rnun 5576 . . . . . . . . . . 11 ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))
3937, 38syl6sseqr 3685 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
40393adantl3 1239 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
41 dff1o3 6181 . . . . . . . . . . . 12 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 ↔ (𝑔:(1...(#‘𝐴))–onto𝐴 ∧ Fun 𝑔))
4241simprbi 479 . . . . . . . . . . 11 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 → Fun 𝑔)
4342adantl 481 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → Fun 𝑔)
44 cnvun 5573 . . . . . . . . . . . . . 14 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) = (𝑔(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))
4544reseq1i 5424 . . . . . . . . . . . . 13 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46 resundir 5446 . . . . . . . . . . . . 13 ((𝑔(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴))
4745, 46eqtri 2673 . . . . . . . . . . . 12 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴))
48 dff1o4 6183 . . . . . . . . . . . . . . . . 17 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 ↔ (𝑔 Fn (1...(#‘𝐴)) ∧ 𝑔 Fn 𝐴))
4948simprbi 479 . . . . . . . . . . . . . . . 16 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴𝑔 Fn 𝐴)
50 fnresdm 6038 . . . . . . . . . . . . . . . 16 (𝑔 Fn 𝐴 → (𝑔𝐴) = 𝑔)
5149, 50syl 17 . . . . . . . . . . . . . . 15 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 → (𝑔𝐴) = 𝑔)
5251adantl 481 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (𝑔𝐴) = 𝑔)
53 simpl3 1086 . . . . . . . . . . . . . . 15 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ¬ 𝑍𝐴)
5414cnveqi 5329 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(#‘𝐴))) × {𝑍})
55 cnvxp 5586 . . . . . . . . . . . . . . . . . 18 ((ℕ ∖ (1...(#‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(#‘𝐴))))
5654, 55eqtri 2673 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(#‘𝐴))))
5756reseq1i 5424 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴)
58 incom 3838 . . . . . . . . . . . . . . . . . 18 (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59 disjsn 4278 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍𝐴)
6059biimpri 218 . . . . . . . . . . . . . . . . . 18 𝑍𝐴 → (𝐴 ∩ {𝑍}) = ∅)
6158, 60syl5eqr 2699 . . . . . . . . . . . . . . . . 17 𝑍𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62 xpdisjres 29537 . . . . . . . . . . . . . . . . 17 (({𝑍} ∩ 𝐴) = ∅ → (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴) = ∅)
6361, 62syl 17 . . . . . . . . . . . . . . . 16 𝑍𝐴 → (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴) = ∅)
6457, 63syl5eq 2697 . . . . . . . . . . . . . . 15 𝑍𝐴 → ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴) = ∅)
6553, 64syl 17 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴) = ∅)
6652, 65uneq12d 3801 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = (𝑔 ∪ ∅))
67 un0 4000 . . . . . . . . . . . . 13 (𝑔 ∪ ∅) = 𝑔
6866, 67syl6eq 2701 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = 𝑔)
6947, 68syl5eq 2697 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = 𝑔)
7069funeqd 5948 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) ↔ Fun 𝑔))
7143, 70mpbird 247 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
72 vex 3234 . . . . . . . . . . 11 𝑔 ∈ V
73 nnex 11064 . . . . . . . . . . . . 13 ℕ ∈ V
74 difexg 4841 . . . . . . . . . . . . 13 (ℕ ∈ V → (ℕ ∖ (1...(#‘𝐴))) ∈ V)
7573, 74ax-mp 5 . . . . . . . . . . . 12 (ℕ ∖ (1...(#‘𝐴))) ∈ V
7675mptex 6527 . . . . . . . . . . 11 (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ∈ V
7772, 76unex 6998 . . . . . . . . . 10 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ∈ V
78 feq1 6064 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79 rneq 5383 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
8079sseq2d 3666 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))))
81 cnveq 5328 . . . . . . . . . . . . 13 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → 𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
82 eqidd 2652 . . . . . . . . . . . . 13 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
8381, 82reseq12d 5429 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝑓𝐴) = ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
8483funeqd 5948 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (Fun (𝑓𝐴) ↔ Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
8578, 80, 843anbi123d 1439 . . . . . . . . . 10 (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 3331 . . . . . . . . 9 (((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 1366 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
8887ex 449 . . . . . . 7 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
899, 10, 88exlimd 2125 . . . . . 6 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → (∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
908, 89mpd 15 . . . . 5 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
91903expia 1286 . . . 4 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
92 nnenom 12819 . . . . . . . 8 ℕ ≈ ω
93 simpl 472 . . . . . . . . 9 ((𝐴 ≈ ω ∧ 𝑍𝑉) → 𝐴 ≈ ω)
9493ensymd 8048 . . . . . . . 8 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ω ≈ 𝐴)
95 entr 8049 . . . . . . . 8 ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
9692, 94, 95sylancr 696 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ℕ ≈ 𝐴)
97 bren 8006 . . . . . . 7 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
9896, 97sylib 208 . . . . . 6 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
99 nfv 1883 . . . . . . 7 𝑓(𝐴 ≈ ω ∧ 𝑍𝑉)
100 simpr 476 . . . . . . . . . 10 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–1-1-onto𝐴)
101 f1of 6175 . . . . . . . . . 10 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
102 ssun1 3809 . . . . . . . . . . 11 𝐴 ⊆ (𝐴 ∪ {𝑍})
103 fss 6094 . . . . . . . . . . 11 ((𝑓:ℕ⟶𝐴𝐴 ⊆ (𝐴 ∪ {𝑍})) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
104102, 103mpan2 707 . . . . . . . . . 10 (𝑓:ℕ⟶𝐴𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
105100, 101, 1043syl 18 . . . . . . . . 9 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
106 f1ofo 6182 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
107 forn 6156 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → ran 𝑓 = 𝐴)
108100, 106, 1073syl 18 . . . . . . . . . 10 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → ran 𝑓 = 𝐴)
10929, 108syl5sseqr 3687 . . . . . . . . 9 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑓)
110 f1ocnv 6187 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:𝐴1-1-onto→ℕ)
111 f1of1 6174 . . . . . . . . . . 11 (𝑓:𝐴1-1-onto→ℕ → 𝑓:𝐴1-1→ℕ)
112100, 110, 1113syl 18 . . . . . . . . . 10 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:𝐴1-1→ℕ)
113 f1ores 6189 . . . . . . . . . . 11 ((𝑓:𝐴1-1→ℕ ∧ 𝐴𝐴) → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
11429, 113mpan2 707 . . . . . . . . . 10 (𝑓:𝐴1-1→ℕ → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
115 f1ofun 6177 . . . . . . . . . 10 ((𝑓𝐴):𝐴1-1-onto→(𝑓𝐴) → Fun (𝑓𝐴))
116112, 114, 1153syl 18 . . . . . . . . 9 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → Fun (𝑓𝐴))
117105, 109, 1163jca 1261 . . . . . . . 8 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
118117ex 449 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → (𝑓:ℕ–1-1-onto𝐴 → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
11999, 118eximd 2123 . . . . . 6 ((𝐴 ≈ ω ∧ 𝑍𝑉) → (∃𝑓 𝑓:ℕ–1-1-onto𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
12098, 119mpd 15 . . . . 5 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
121120a1d 25 . . . 4 ((𝐴 ≈ ω ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
12291, 121jaoian 841 . . 3 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
1231223impia 1280 . 2 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
1241, 123syl3an1b 1402 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 196  wo 382  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   class class class wbr 4685  cmpt 4762   × cxp 5141  ccnv 5142  ran crn 5144  cres 5145  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  ωcom 7107  cen 7994  cdom 7995  csdm 7996  Fincfn 7997  1c1 9975  cn 11058  ...cfz 12364  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158
This theorem is referenced by:  carsggect  30508
  Copyright terms: Public domain W3C validator