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Theorem padct 31931
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 8974 . 2 (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2 nfv 1917 . . . . . 6 𝑔(𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴)
3 nfv 1917 . . . . . 6 𝑔𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))
4 isfinite2 9297 . . . . . . . . . 10 (𝐴 ≺ ω → 𝐴 ∈ Fin)
5 isfinite4 14318 . . . . . . . . . 10 (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴)
64, 5sylib 217 . . . . . . . . 9 (𝐴 ≺ ω → (1...(♯‘𝐴)) ≈ 𝐴)
76adantr 481 . . . . . . . 8 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (1...(♯‘𝐴)) ≈ 𝐴)
8 bren 8945 . . . . . . . 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 1132 . . . . . 6 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
11 f1of 6830 . . . . . . . . . . . 12 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴𝑔:(1...(♯‘𝐴))⟶𝐴)
1211adantl 482 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑔:(1...(♯‘𝐴))⟶𝐴)
13 fconstmpt 5736 . . . . . . . . . . . . 13 ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)
1413eqcomi 2741 . . . . . . . . . . . 12 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
15 simplr 767 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑍𝑉)
16 fconst2g 7200 . . . . . . . . . . . . 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 257 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(♯‘𝐴)))⟶{𝑍})
19 disjdif 4470 . . . . . . . . . . . 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 6750 . . . . . . . . . . 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 836 . . . . . . . . . 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 13528 . . . . . . . . . . . 12 (1...(♯‘𝐴)) ⊆ ℕ
24 undif 4480 . . . . . . . . . . . 12 ((1...(♯‘𝐴)) ⊆ ℕ ↔ ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ)
2523, 24mpbi 229 . . . . . . . . . . 11 ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ
2625feq2i 6706 . . . . . . . . . 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 1168 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
29 ssid 4003 . . . . . . . . . . . . 13 𝐴𝐴
30 simpr 485 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴)
31 f1ofo 6837 . . . . . . . . . . . . . 14 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴𝑔:(1...(♯‘𝐴))–onto𝐴)
32 forn 6805 . . . . . . . . . . . . . 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 4034 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑔)
3534orcd 871 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝐴 ⊆ ran 𝑔𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
36 ssun 4188 . . . . . . . . . . 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 6142 . . . . . . . . . 10 ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
3937, 38sseqtrrdi 4032 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
40393adantl3 1168 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
41 dff1o3 6836 . . . . . . . . . . 11 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴 ↔ (𝑔:(1...(♯‘𝐴))–onto𝐴 ∧ Fun 𝑔))
4241simprbi 497 . . . . . . . . . 10 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴 → Fun 𝑔)
4342adantl 482 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → Fun 𝑔)
44 cnvun 6139 . . . . . . . . . . . . 13 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
4544reseq1i 5975 . . . . . . . . . . . 12 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46 resundir 5994 . . . . . . . . . . . 12 ((𝑔(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
4745, 46eqtri 2760 . . . . . . . . . . 11 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
48 dff1o4 6838 . . . . . . . . . . . . . . . 16 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴 ↔ (𝑔 Fn (1...(♯‘𝐴)) ∧ 𝑔 Fn 𝐴))
4948simprbi 497 . . . . . . . . . . . . . . 15 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴𝑔 Fn 𝐴)
50 fnresdm 6666 . . . . . . . . . . . . . . 15 (𝑔 Fn 𝐴 → (𝑔𝐴) = 𝑔)
5149, 50syl 17 . . . . . . . . . . . . . 14 (𝑔:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝑔𝐴) = 𝑔)
5251adantl 482 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝑔𝐴) = 𝑔)
53 simpl3 1193 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ¬ 𝑍𝐴)
5414cnveqi 5872 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
55 cnvxp 6153 . . . . . . . . . . . . . . . . 17 ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
5654, 55eqtri 2760 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
5756reseq1i 5975 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴)
58 incom 4200 . . . . . . . . . . . . . . . . 17 (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59 disjsn 4714 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍𝐴)
6059biimpri 227 . . . . . . . . . . . . . . . . 17 𝑍𝐴 → (𝐴 ∩ {𝑍}) = ∅)
6158, 60eqtr3id 2786 . . . . . . . . . . . . . . . 16 𝑍𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62 xpdisjres 31816 . . . . . . . . . . . . . . . 16 (({𝑍} ∩ 𝐴) = ∅ → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
6361, 62syl 17 . . . . . . . . . . . . . . 15 𝑍𝐴 → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
6457, 63eqtrid 2784 . . . . . . . . . . . . . 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 4389 . . . . . . . . . . . 12 (𝑔 ∪ ∅) = 𝑔
6866, 67eqtrdi 2788 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = 𝑔)
6947, 68eqtrid 2784 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = 𝑔)
7069funeqd 6567 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → (Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) ↔ Fun 𝑔))
7143, 70mpbird 256 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
72 vex 3478 . . . . . . . . . 10 𝑔 ∈ V
73 nnex 12214 . . . . . . . . . . . 12 ℕ ∈ V
74 difexg 5326 . . . . . . . . . . . 12 (ℕ ∈ V → (ℕ ∖ (1...(♯‘𝐴))) ∈ V)
7573, 74ax-mp 5 . . . . . . . . . . 11 (ℕ ∖ (1...(♯‘𝐴))) ∈ V
7675mptex 7221 . . . . . . . . . 10 (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ∈ V
7772, 76unex 7729 . . . . . . . . 9 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ∈ V
78 feq1 6695 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79 rneq 5933 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
8079sseq2d 4013 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))))
81 cnveq 5871 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
82 eqidd 2733 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
8381, 82reseq12d 5980 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓𝐴) = ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
8483funeqd 6567 . . . . . . . . . 10 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (Fun (𝑓𝐴) ↔ Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
8578, 80, 843anbi123d 1436 . . . . . . . . 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 3596 . . . . . . . 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 1371 . . . . . . 7 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
8887ex 413 . . . . . 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 2212 . . . . 5 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
90893expia 1121 . . . 4 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
91 nnenom 13941 . . . . . . . 8 ℕ ≈ ω
92 simpl 483 . . . . . . . . 9 ((𝐴 ≈ ω ∧ 𝑍𝑉) → 𝐴 ≈ ω)
9392ensymd 8997 . . . . . . . 8 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ω ≈ 𝐴)
94 entr 8998 . . . . . . . 8 ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
9591, 93, 94sylancr 587 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ℕ ≈ 𝐴)
96 bren 8945 . . . . . . 7 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
9795, 96sylib 217 . . . . . 6 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
98 nfv 1917 . . . . . . 7 𝑓(𝐴 ≈ ω ∧ 𝑍𝑉)
99 simpr 485 . . . . . . . . . 10 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–1-1-onto𝐴)
100 f1of 6830 . . . . . . . . . 10 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
101 ssun1 4171 . . . . . . . . . . 11 𝐴 ⊆ (𝐴 ∪ {𝑍})
102 fss 6731 . . . . . . . . . . 11 ((𝑓:ℕ⟶𝐴𝐴 ⊆ (𝐴 ∪ {𝑍})) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
103101, 102mpan2 689 . . . . . . . . . 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 6837 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
106 forn 6805 . . . . . . . . . . 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 4034 . . . . . . . . 9 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑓)
109 f1ocnv 6842 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:𝐴1-1-onto→ℕ)
110 f1of1 6829 . . . . . . . . . . 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 6844 . . . . . . . . . . 11 ((𝑓:𝐴1-1→ℕ ∧ 𝐴𝐴) → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
11329, 112mpan2 689 . . . . . . . . . 10 (𝑓:𝐴1-1→ℕ → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
114 f1ofun 6832 . . . . . . . . . 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 1128 . . . . . . . 8 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
117116ex 413 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → (𝑓:ℕ–1-1-onto𝐴 → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
11898, 117eximd 2209 . . . . . 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 1117 . 2 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
1231, 122syl3an1b 1403 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 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321  {csn 4627   class class class wbr 5147  cmpt 5230   × cxp 5673  ccnv 5674  ran crn 5676  cres 5677  cima 5678  Fun wfun 6534   Fn wfn 6535  wf 6536  1-1wf1 6537  ontowfo 6538  1-1-ontowf1o 6539  cfv 6540  (class class class)co 7405  ωcom 7851  cen 8932  cdom 8933  csdm 8934  Fincfn 8935  1c1 11107  cn 12208  ...cfz 13480  chash 14286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287
This theorem is referenced by:  carsggect  33305
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