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Theorem padct 32736
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 9020 . 2 (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2 nfv 1911 . . . . . 6 𝑔(𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴)
3 nfv 1911 . . . . . 6 𝑔𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))
4 isfinite2 9331 . . . . . . . . . 10 (𝐴 ≺ ω → 𝐴 ∈ Fin)
5 isfinite4 14397 . . . . . . . . . 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 8993 . . . . . . . 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 1131 . . . . . 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 5750 . . . . . . . . . . . . 13 ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)
1413eqcomi 2743 . . . . . . . . . . . 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 7222 . . . . . . . . . . . . 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 4477 . . . . . . . . . . . 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 13591 . . . . . . . . . . . 12 (1...(♯‘𝐴)) ⊆ ℕ
24 undif 4487 . . . . . . . . . . . 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 1167 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
29 ssid 4017 . . . . . . . . . . . . 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 4048 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑔)
3534orcd 873 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝐴 ⊆ ran 𝑔𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
36 ssun 4204 . . . . . . . . . . 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 6167 . . . . . . . . . 10 ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
3937, 38sseqtrrdi 4046 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
40393adantl3 1167 . . . . . . . 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 6164 . . . . . . . . . . . . 13 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
4544reseq1i 5995 . . . . . . . . . . . 12 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46 resundir 6014 . . . . . . . . . . . 12 ((𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
4745, 46eqtri 2762 . . . . . . . . . . 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 1192 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ¬ 𝑍𝐴)
5414cnveqi 5887 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
55 cnvxp 6178 . . . . . . . . . . . . . . . . 17 ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
5654, 55eqtri 2762 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
5756reseq1i 5995 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴)
58 incom 4216 . . . . . . . . . . . . . . . . 17 (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59 disjsn 4715 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍𝐴)
6059biimpri 228 . . . . . . . . . . . . . . . . 17 𝑍𝐴 → (𝐴 ∩ {𝑍}) = ∅)
6158, 60eqtr3id 2788 . . . . . . . . . . . . . . . 16 𝑍𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62 xpdisjres 32617 . . . . . . . . . . . . . . . 16 (({𝑍} ∩ 𝐴) = ∅ → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
6361, 62syl 17 . . . . . . . . . . . . . . 15 𝑍𝐴 → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
6457, 63eqtrid 2786 . . . . . . . . . . . . . 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 4178 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = (𝑔 ∪ ∅))
67 un0 4399 . . . . . . . . . . . 12 (𝑔 ∪ ∅) = 𝑔
6866, 67eqtrdi 2790 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = 𝑔)
6947, 68eqtrid 2786 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = 𝑔)
7069funeqd 6589 . . . . . . . . 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 3481 . . . . . . . . . 10 𝑔 ∈ V
73 nnex 12269 . . . . . . . . . . . 12 ℕ ∈ V
74 difexg 5334 . . . . . . . . . . . 12 (ℕ ∈ V → (ℕ ∖ (1...(♯‘𝐴))) ∈ V)
7573, 74ax-mp 5 . . . . . . . . . . 11 (ℕ ∖ (1...(♯‘𝐴))) ∈ V
7675mptex 7242 . . . . . . . . . 10 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ∈ V
7772, 76unex 7762 . . . . . . . . 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 5949 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
8079sseq2d 4027 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))))
81 cnveq 5886 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
82 eqidd 2735 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
8381, 82reseq12d 6000 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓𝐴) = ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
8483funeqd 6589 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (Fun (𝑓𝐴) ↔ Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
8578, 80, 843anbi123d 1435 . . . . . . . . 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 3605 . . . . . . . 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 1370 . . . . . . 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 2216 . . . . 5 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
90893expia 1120 . . . 4 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
91 nnenom 14017 . . . . . . . 8 ℕ ≈ ω
92 simpl 482 . . . . . . . . 9 ((𝐴 ≈ ω ∧ 𝑍𝑉) → 𝐴 ≈ ω)
9392ensymd 9043 . . . . . . . 8 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ω ≈ 𝐴)
94 entr 9044 . . . . . . . 8 ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
9591, 93, 94sylancr 587 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ℕ ≈ 𝐴)
96 bren 8993 . . . . . . 7 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
9795, 96sylib 218 . . . . . 6 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
98 nfv 1911 . . . . . . 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 4187 . . . . . . . . . . 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 4048 . . . . . . . . 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 1127 . . . . . . . 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 2213 . . . . . 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 958 . . 3 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
1221213impia 1116 . 2 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
1231, 122syl3an1b 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 206  wa 395  wo 847  w3a 1086   = wceq 1536  wex 1775  wcel 2105  Vcvv 3477  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  {csn 4630   class class class wbr 5147  cmpt 5230   × cxp 5686  ccnv 5687  ran crn 5689  cres 5690  cima 5691  Fun wfun 6556   Fn wfn 6557  wf 6558  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  ωcom 7886  cen 8980  cdom 8981  csdm 8982  Fincfn 8983  1c1 11153  cn 12263  ...cfz 13543  chash 14365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366
This theorem is referenced by:  carsggect  34299
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