Theorem padct 29497

Description: Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.)

Assertion

Ref	Expression
padct	⊢ ((𝐴 ≼ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))

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.

Step	Hyp	Ref	Expression
1		brdom2 7985	. 2 ⊢ (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2		isfinite2 8218	. . . . . . . . . 10 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin)
3		isfinite4 13153	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin ↔ (1...(#‘𝐴)) ≈ 𝐴)
4	2, 3	sylib 208	. . . . . . . . 9 ⊢ (𝐴 ≺ ω → (1...(#‘𝐴)) ≈ 𝐴)
5	4	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (1...(#‘𝐴)) ≈ 𝐴)
6		bren 7964	. . . . . . . 8 ⊢ ((1...(#‘𝐴)) ≈ 𝐴 ↔ ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴)
7	5, 6	sylib 208	. . . . . . 7 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴)
8	7	3adant3 1081	. . . . . 6 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴)
9		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑔(𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴)
10		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑔∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))
11		f1of 6137	. . . . . . . . . . . . 13 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(#‘𝐴))⟶𝐴)
12	11	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(#‘𝐴))⟶𝐴)
13		fconstmpt 5163	. . . . . . . . . . . . . 14 ⊢ ((ℕ ∖ (1...(#‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)
14	13	eqcomi 2631	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(#‘𝐴))) × {𝑍})
15		simplr 792	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝑍 ∈ 𝑉)
16		fconst2g 6468	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ 𝑉 → ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(#‘𝐴)))⟶{𝑍} ↔ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(#‘𝐴))) × {𝑍})))
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → ((𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(#‘𝐴)))⟶{𝑍} ↔ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(#‘𝐴))) × {𝑍})))
18	14, 17	mpbiri 248	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(#‘𝐴)))⟶{𝑍})
19		disjdif 4040	. . . . . . . . . . . . 13 ⊢ ((1...(#‘𝐴)) ∩ (ℕ ∖ (1...(#‘𝐴)))) = ∅
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → ((1...(#‘𝐴)) ∩ (ℕ ∖ (1...(#‘𝐴)))) = ∅)
21		fun 6066	. . . . . . . . . . . 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}))
22	12, 18, 20, 21	syl21anc 1325	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴))))⟶(𝐴 ∪ {𝑍}))
23		fz1ssnn 12372	. . . . . . . . . . . . 13 ⊢ (1...(#‘𝐴)) ⊆ ℕ
24		undif 4049	. . . . . . . . . . . . 13 ⊢ ((1...(#‘𝐴)) ⊆ ℕ ↔ ((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴)))) = ℕ)
25	23, 24	mpbi 220	. . . . . . . . . . . 12 ⊢ ((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴)))) = ℕ
26	25	feq2i 6037	. . . . . . . . . . 11 ⊢ ((𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴))))⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
27	22, 26	sylib 208	. . . . . . . . . 10 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
28	27	3adantl3 1219	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
29		ssid 3624	. . . . . . . . . . . . . 14 ⊢ 𝐴 ⊆ 𝐴
30		simpr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴)
31		f1ofo 6144	. . . . . . . . . . . . . . 15 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(#‘𝐴))–onto→𝐴)
32		forn 6118	. . . . . . . . . . . . . . 15 ⊢ (𝑔:(1...(#‘𝐴))–onto→𝐴 → ran 𝑔 = 𝐴)
33	30, 31, 32	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → ran 𝑔 = 𝐴)
34	29, 33	syl5sseqr 3654	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran 𝑔)
35	34	orcd 407	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝐴 ⊆ ran 𝑔 ∨ 𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
36		ssun 3792	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ran 𝑔 ∨ 𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → 𝐴 ⊆ (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
38		rnun 5541	. . . . . . . . . . 11 ⊢ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))
39	37, 38	syl6sseqr 3652	. . . . . . . . . 10 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
40	39	3adantl3 1219	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
41		dff1o3 6143	. . . . . . . . . . . 12 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 ↔ (𝑔:(1...(#‘𝐴))–onto→𝐴 ∧ Fun ◡𝑔))
42	41	simprbi 480	. . . . . . . . . . 11 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → Fun ◡𝑔)
43	42	adantl 482	. . . . . . . . . 10 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → Fun ◡𝑔)
44		cnvun 5538	. . . . . . . . . . . . . 14 ⊢ ◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) = (◡𝑔 ∪ ◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))
45	44	reseq1i 5392	. . . . . . . . . . . . 13 ⊢ (◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((◡𝑔 ∪ ◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46		resundir 5411	. . . . . . . . . . . . 13 ⊢ ((◡𝑔 ∪ ◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((◡𝑔 ↾ 𝐴) ∪ (◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴))
47	45, 46	eqtri 2644	. . . . . . . . . . . 12 ⊢ (◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((◡𝑔 ↾ 𝐴) ∪ (◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴))
48		dff1o4 6145	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 ↔ (𝑔 Fn (1...(#‘𝐴)) ∧ ◡𝑔 Fn 𝐴))
49	48	simprbi 480	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → ◡𝑔 Fn 𝐴)
50		fnresdm 6000	. . . . . . . . . . . . . . . 16 ⊢ (◡𝑔 Fn 𝐴 → (◡𝑔 ↾ 𝐴) = ◡𝑔)
51	49, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → (◡𝑔 ↾ 𝐴) = ◡𝑔)
52	51	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (◡𝑔 ↾ 𝐴) = ◡𝑔)
53		simpl3 1066	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → ¬ 𝑍 ∈ 𝐴)
54	14	cnveqi 5297	. . . . . . . . . . . . . . . . . 18 ⊢ ◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ◡((ℕ ∖ (1...(#‘𝐴))) × {𝑍})
55		cnvxp 5551	. . . . . . . . . . . . . . . . . 18 ⊢ ◡((ℕ ∖ (1...(#‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(#‘𝐴))))
56	54, 55	eqtri 2644	. . . . . . . . . . . . . . . . 17 ⊢ ◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(#‘𝐴))))
57	56	reseq1i 5392	. . . . . . . . . . . . . . . 16 ⊢ (◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴)
58		incom 3805	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59		disjsn 4246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍 ∈ 𝐴)
60	59	biimpri 218	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑍 ∈ 𝐴 → (𝐴 ∩ {𝑍}) = ∅)
61	58, 60	syl5eqr 2670	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑍 ∈ 𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62		xpdisjres 29411	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑍} ∩ 𝐴) = ∅ → (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴) = ∅)
63	61, 62	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑍 ∈ 𝐴 → (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴) = ∅)
64	57, 63	syl5eq 2668	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑍 ∈ 𝐴 → (◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴) = ∅)
65	53, 64	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴) = ∅)
66	52, 65	uneq12d 3768	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → ((◡𝑔 ↾ 𝐴) ∪ (◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = (◡𝑔 ∪ ∅))
67		un0 3967	. . . . . . . . . . . . 13 ⊢ (◡𝑔 ∪ ∅) = ◡𝑔
68	66, 67	syl6eq 2672	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → ((◡𝑔 ↾ 𝐴) ∪ (◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = ◡𝑔)
69	47, 68	syl5eq 2668	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ◡𝑔)
70	69	funeqd 5910	. . . . . . . . . 10 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (Fun (◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) ↔ Fun ◡𝑔))
71	43, 70	mpbird 247	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → Fun (◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
72		vex 3203	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
73		nnex 11026	. . . . . . . . . . . . 13 ⊢ ℕ ∈ V
74		difexg 4808	. . . . . . . . . . . . 13 ⊢ (ℕ ∈ V → (ℕ ∖ (1...(#‘𝐴))) ∈ V)
75	73, 74	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℕ ∖ (1...(#‘𝐴))) ∈ V
76	75	mptex 6486	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ∈ V
77	72, 76	unex 6956	. . . . . . . . . 10 ⊢ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ∈ V
78		feq1 6026	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79		rneq 5351	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
80	79	sseq2d 3633	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓 ↔ 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))))
81		cnveq 5296	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → ◡𝑓 = ◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
82		eqidd 2623	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
83	81, 82	reseq12d 5397	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (◡𝑓 ↾ 𝐴) = (◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
84	83	funeqd 5910	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (Fun (◡𝑓 ↾ 𝐴) ↔ Fun (◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
85	78, 80, 84	3anbi123d 1399	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → ((𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)) ↔ ((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))))
86	77, 85	spcev 3300	. . . . . . . . 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 (◡𝑓 ↾ 𝐴)))
87	28, 40, 71, 86	syl3anc 1326	. . . . . . . 8 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
88	87	ex 450	. . . . . . 7 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))))
89	9, 10, 88	exlimd 2087	. . . . . 6 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → (∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))))
90	8, 89	mpd 15	. . . . 5 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
91	90	3expia 1267	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))))
92		nnenom 12779	. . . . . . . 8 ⊢ ℕ ≈ ω
93		simpl 473	. . . . . . . . 9 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → 𝐴 ≈ ω)
94	93	ensymd 8007	. . . . . . . 8 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ω ≈ 𝐴)
95		entr 8008	. . . . . . . 8 ⊢ ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
96	92, 94, 95	sylancr 695	. . . . . . 7 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ℕ ≈ 𝐴)
97		bren 7964	. . . . . . 7 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
98	96, 97	sylib 208	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
99		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑓(𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉)
100		simpr 477	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ–1-1-onto→𝐴)
101		f1of 6137	. . . . . . . . . 10 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ⟶𝐴)
102		ssun1 3776	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑍})
103		fss 6056	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶𝐴 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑍})) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
104	102, 103	mpan2 707	. . . . . . . . . 10 ⊢ (𝑓:ℕ⟶𝐴 → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
105	100, 101, 104	3syl 18	. . . . . . . . 9 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
106		f1ofo 6144	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ–onto→𝐴)
107		forn 6118	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → ran 𝑓 = 𝐴)
108	100, 106, 107	3syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran 𝑓 = 𝐴)
109	29, 108	syl5sseqr 3654	. . . . . . . . 9 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝐴 ⊆ ran 𝑓)
110		f1ocnv 6149	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→ℕ)
111		f1of1 6136	. . . . . . . . . . 11 ⊢ (◡𝑓:𝐴–1-1-onto→ℕ → ◡𝑓:𝐴–1-1→ℕ)
112	100, 110, 111	3syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → ◡𝑓:𝐴–1-1→ℕ)
113		f1ores 6151	. . . . . . . . . . 11 ⊢ ((◡𝑓:𝐴–1-1→ℕ ∧ 𝐴 ⊆ 𝐴) → (◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(◡𝑓 “ 𝐴))
114	29, 113	mpan2 707	. . . . . . . . . 10 ⊢ (◡𝑓:𝐴–1-1→ℕ → (◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(◡𝑓 “ 𝐴))
115		f1ofun 6139	. . . . . . . . . 10 ⊢ ((◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(◡𝑓 “ 𝐴) → Fun (◡𝑓 ↾ 𝐴))
116	112, 114, 115	3syl 18	. . . . . . . . 9 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → Fun (◡𝑓 ↾ 𝐴))
117	105, 109, 116	3jca 1242	. . . . . . . 8 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
118	117	ex 450	. . . . . . 7 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → (𝑓:ℕ–1-1-onto→𝐴 → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))))
119	99, 118	eximd 2085	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → (∃𝑓 𝑓:ℕ–1-1-onto→𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))))
120	98, 119	mpd 15	. . . . 5 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
121	120	a1d 25	. . . 4 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))))
122	91, 121	jaoian 824	. . 3 ⊢ (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))))
123	122	3impia 1261	. 2 ⊢ (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
124	1, 123	syl3an1b 1362	1 ⊢ ((𝐴 ≼ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  ωcom 7065   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  1c1 9937  ℕcn 11020  ...cfz 12326  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  carsggect  30380