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Theorem padct 29340
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 7929 . 2 (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2 isfinite2 8162 . . . . . . . . . 10 (𝐴 ≺ ω → 𝐴 ∈ Fin)
3 isfinite4 13093 . . . . . . . . . 10 (𝐴 ∈ Fin ↔ (1...(#‘𝐴)) ≈ 𝐴)
42, 3sylib 208 . . . . . . . . 9 (𝐴 ≺ ω → (1...(#‘𝐴)) ≈ 𝐴)
54adantr 481 . . . . . . . 8 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (1...(#‘𝐴)) ≈ 𝐴)
6 bren 7908 . . . . . . . 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 1079 . . . . . 6 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto𝐴)
9 nfv 1840 . . . . . . 7 𝑔(𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴)
10 nfv 1840 . . . . . . 7 𝑔𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))
11 f1of 6094 . . . . . . . . . . . . 13 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴𝑔:(1...(#‘𝐴))⟶𝐴)
1211adantl 482 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝑔:(1...(#‘𝐴))⟶𝐴)
13 fconstmpt 5123 . . . . . . . . . . . . . 14 ((ℕ ∖ (1...(#‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)
1413eqcomi 2630 . . . . . . . . . . . . 13 (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(#‘𝐴))) × {𝑍})
15 simplr 791 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝑍𝑉)
16 fconst2g 6422 . . . . . . . . . . . . . 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 4012 . . . . . . . . . . . . 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 6023 . . . . . . . . . . . 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 1322 . . . . . . . . . . 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 12314 . . . . . . . . . . . . 13 (1...(#‘𝐴)) ⊆ ℕ
24 undif 4021 . . . . . . . . . . . . 13 ((1...(#‘𝐴)) ⊆ ℕ ↔ ((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴)))) = ℕ)
2523, 24mpbi 220 . . . . . . . . . . . 12 ((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴)))) = ℕ
2625feq2i 5994 . . . . . . . . . . 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 1217 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
29 ssid 3603 . . . . . . . . . . . . . 14 𝐴𝐴
30 simpr 477 . . . . . . . . . . . . . . 15 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝑔:(1...(#‘𝐴))–1-1-onto𝐴)
31 f1ofo 6101 . . . . . . . . . . . . . . 15 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴𝑔:(1...(#‘𝐴))–onto𝐴)
32 forn 6075 . . . . . . . . . . . . . . 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 3633 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑔)
3534orcd 407 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (𝐴 ⊆ ran 𝑔𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
36 ssun 3770 . . . . . . . . . . . 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 5500 . . . . . . . . . . 11 ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))
3937, 38syl6sseqr 3631 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
40393adantl3 1217 . . . . . . . . 9 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
41 dff1o3 6100 . . . . . . . . . . . 12 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 ↔ (𝑔:(1...(#‘𝐴))–onto𝐴 ∧ Fun 𝑔))
4241simprbi 480 . . . . . . . . . . 11 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 → Fun 𝑔)
4342adantl 482 . . . . . . . . . 10 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → Fun 𝑔)
44 cnvun 5497 . . . . . . . . . . . . . 14 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) = (𝑔(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))
4544reseq1i 5352 . . . . . . . . . . . . 13 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46 resundir 5370 . . . . . . . . . . . . 13 ((𝑔(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴))
4745, 46eqtri 2643 . . . . . . . . . . . 12 ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴))
48 dff1o4 6102 . . . . . . . . . . . . . . . . 17 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 ↔ (𝑔 Fn (1...(#‘𝐴)) ∧ 𝑔 Fn 𝐴))
4948simprbi 480 . . . . . . . . . . . . . . . 16 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴𝑔 Fn 𝐴)
50 fnresdm 5958 . . . . . . . . . . . . . . . 16 (𝑔 Fn 𝐴 → (𝑔𝐴) = 𝑔)
5149, 50syl 17 . . . . . . . . . . . . . . 15 (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 → (𝑔𝐴) = 𝑔)
5251adantl 482 . . . . . . . . . . . . . 14 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → (𝑔𝐴) = 𝑔)
53 simpl3 1064 . . . . . . . . . . . . . . 15 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ¬ 𝑍𝐴)
5414cnveqi 5257 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(#‘𝐴))) × {𝑍})
55 cnvxp 5510 . . . . . . . . . . . . . . . . . 18 ((ℕ ∖ (1...(#‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(#‘𝐴))))
5654, 55eqtri 2643 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(#‘𝐴))))
5756reseq1i 5352 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴)
58 incom 3783 . . . . . . . . . . . . . . . . . 18 (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59 disjsn 4216 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍𝐴)
6059biimpri 218 . . . . . . . . . . . . . . . . . 18 𝑍𝐴 → (𝐴 ∩ {𝑍}) = ∅)
6158, 60syl5eqr 2669 . . . . . . . . . . . . . . . . 17 𝑍𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62 xpdisjres 29256 . . . . . . . . . . . . . . . . 17 (({𝑍} ∩ 𝐴) = ∅ → (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴) = ∅)
6361, 62syl 17 . . . . . . . . . . . . . . . 16 𝑍𝐴 → (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴) = ∅)
6457, 63syl5eq 2667 . . . . . . . . . . . . . . 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 3746 . . . . . . . . . . . . 13 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = (𝑔 ∪ ∅))
67 un0 3939 . . . . . . . . . . . . 13 (𝑔 ∪ ∅) = 𝑔
6866, 67syl6eq 2671 . . . . . . . . . . . 12 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ((𝑔𝐴) ∪ ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = 𝑔)
6947, 68syl5eq 2667 . . . . . . . . . . 11 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = 𝑔)
7069funeqd 5869 . . . . . . . . . 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 3189 . . . . . . . . . . 11 𝑔 ∈ V
73 nnex 10970 . . . . . . . . . . . . 13 ℕ ∈ V
74 difexg 4768 . . . . . . . . . . . . 13 (ℕ ∈ V → (ℕ ∖ (1...(#‘𝐴))) ∈ V)
7573, 74ax-mp 5 . . . . . . . . . . . 12 (ℕ ∖ (1...(#‘𝐴))) ∈ V
7675mptex 6440 . . . . . . . . . . 11 (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ∈ V
7772, 76unex 6909 . . . . . . . . . 10 (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ∈ V
78 feq1 5983 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79 rneq 5311 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
8079sseq2d 3612 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))))
81 cnveq 5256 . . . . . . . . . . . . 13 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → 𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
82 eqidd 2622 . . . . . . . . . . . . 13 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
8381, 82reseq12d 5357 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝑓𝐴) = ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
8483funeqd 5869 . . . . . . . . . . 11 (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (Fun (𝑓𝐴) ↔ Fun ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
8578, 80, 843anbi123d 1396 . . . . . . . . . 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 3286 . . . . . . . . 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 1323 . . . . . . . 8 (((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
8887ex 450 . . . . . . 7 ((𝐴 ≺ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → (𝑔:(1...(#‘𝐴))–1-1-onto𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
899, 10, 88exlimd 2085 . . . . . 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 1264 . . . 4 ((𝐴 ≺ ω ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
92 nnenom 12719 . . . . . . . 8 ℕ ≈ ω
93 simpl 473 . . . . . . . . 9 ((𝐴 ≈ ω ∧ 𝑍𝑉) → 𝐴 ≈ ω)
9493ensymd 7951 . . . . . . . 8 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ω ≈ 𝐴)
95 entr 7952 . . . . . . . 8 ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
9692, 94, 95sylancr 694 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ℕ ≈ 𝐴)
97 bren 7908 . . . . . . 7 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
9896, 97sylib 208 . . . . . 6 ((𝐴 ≈ ω ∧ 𝑍𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
99 nfv 1840 . . . . . . 7 𝑓(𝐴 ≈ ω ∧ 𝑍𝑉)
100 simpr 477 . . . . . . . . . 10 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–1-1-onto𝐴)
101 f1of 6094 . . . . . . . . . 10 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
102 ssun1 3754 . . . . . . . . . . 11 𝐴 ⊆ (𝐴 ∪ {𝑍})
103 fss 6013 . . . . . . . . . . 11 ((𝑓:ℕ⟶𝐴𝐴 ⊆ (𝐴 ∪ {𝑍})) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
104102, 103mpan2 706 . . . . . . . . . 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 6101 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
107 forn 6075 . . . . . . . . . . 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 3633 . . . . . . . . 9 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝐴 ⊆ ran 𝑓)
110 f1ocnv 6106 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto𝐴𝑓:𝐴1-1-onto→ℕ)
111 f1of1 6093 . . . . . . . . . . 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 6108 . . . . . . . . . . 11 ((𝑓:𝐴1-1→ℕ ∧ 𝐴𝐴) → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
11429, 113mpan2 706 . . . . . . . . . 10 (𝑓:𝐴1-1→ℕ → (𝑓𝐴):𝐴1-1-onto→(𝑓𝐴))
115 f1ofun 6096 . . . . . . . . . 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 1240 . . . . . . . 8 (((𝐴 ≈ ω ∧ 𝑍𝑉) ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
118117ex 450 . . . . . . 7 ((𝐴 ≈ ω ∧ 𝑍𝑉) → (𝑓:ℕ–1-1-onto𝐴 → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
11999, 118eximd 2083 . . . . . 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 823 . . 3 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉) → (¬ 𝑍𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))))
1231223impia 1258 . 2 (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
1241, 123syl3an1b 1359 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 383  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  {csn 4148   class class class wbr 4613  cmpt 4673   × cxp 5072  ccnv 5073  ran crn 5075  cres 5076  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  ωcom 7012  cen 7896  cdom 7897  csdm 7898  Fincfn 7899  1c1 9881  cn 10964  ...cfz 12268  #chash 13057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058
This theorem is referenced by:  carsggect  30161
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