padct - Mathbox for Thierry Arnoux

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Theorem padct 28885

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 7871	. 2 ⊢ (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2		isfinite2 8103	. . . . . . . . . 10 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin)
3		isfinite4 13014	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin ↔ (1...(#‘𝐴)) ≈ 𝐴)
4	2, 3	sylib 207	. . . . . . . . 9 ⊢ (𝐴 ≺ ω → (1...(#‘𝐴)) ≈ 𝐴)
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (1...(#‘𝐴)) ≈ 𝐴)
6		bren 7850	. . . . . . . 8 ⊢ ((1...(#‘𝐴)) ≈ 𝐴 ↔ ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴)
7	5, 6	sylib 207	. . . . . . 7 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴)
8	7	3adant3 1074	. . . . . 6 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑔 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴)
9		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑔(𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴)
10		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑔∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))
11		f1of 6050	. . . . . . . . . . . . 13 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(#‘𝐴))⟶𝐴)
12	11	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(#‘𝐴))⟶𝐴)
13		fconstmpt 5085	. . . . . . . . . . . . . 14 ⊢ ((ℕ ∖ (1...(#‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)
14	13	eqcomi 2619	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(#‘𝐴))) × {𝑍})
15		simplr 788	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝑍 ∈ 𝑉)
16		fconst2g 6373	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(#‘𝐴)))⟶{𝑍})
19		disjdif 3992	. . . . . . . . . . . . 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 5979	. . . . . . . . . . . 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 1317	. . . . . . . . . . 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 12243	. . . . . . . . . . . . 13 ⊢ (1...(#‘𝐴)) ⊆ ℕ
24		undif 4001	. . . . . . . . . . . . 13 ⊢ ((1...(#‘𝐴)) ⊆ ℕ ↔ ((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴)))) = ℕ)
25	23, 24	mpbi 219	. . . . . . . . . . . 12 ⊢ ((1...(#‘𝐴)) ∪ (ℕ ∖ (1...(#‘𝐴)))) = ℕ
26	25	feq2i 5950	. . . . . . . . . . 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 207	. . . . . . . . . 10 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
28	27	3adantl3 1212	. . . . . . . . 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 3587	. . . . . . . . . . . . . 14 ⊢ 𝐴 ⊆ 𝐴
30		simpr 476	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴)
31		f1ofo 6057	. . . . . . . . . . . . . . 15 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(#‘𝐴))–onto→𝐴)
32		forn 6031	. . . . . . . . . . . . . . 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 3617	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran 𝑔)
35	34	orcd 406	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝐴 ⊆ ran 𝑔 ∨ 𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
36		ssun 3754	. . . . . . . . . . . 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 5460	. . . . . . . . . . 11 ⊢ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))
39	37, 38	syl6sseqr 3615	. . . . . . . . . 10 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
40	39	3adantl3 1212	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
41		dff1o3 6056	. . . . . . . . . . . 12 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 ↔ (𝑔:(1...(#‘𝐴))–onto→𝐴 ∧ Fun ^◡𝑔))
42	41	simprbi 479	. . . . . . . . . . 11 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → Fun ^◡𝑔)
43	42	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → Fun ^◡𝑔)
44		cnvun 5457	. . . . . . . . . . . . . 14 ⊢ ^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) = (^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))
45	44	reseq1i 5313	. . . . . . . . . . . . 13 ⊢ (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46		resundir 5331	. . . . . . . . . . . . 13 ⊢ ((^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ↾ 𝐴) ∪ (^◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴))
47	45, 46	eqtri 2632	. . . . . . . . . . . 12 ⊢ (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ↾ 𝐴) ∪ (^◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴))
48		dff1o4 6058	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 ↔ (𝑔 Fn (1...(#‘𝐴)) ∧ ^◡𝑔 Fn 𝐴))
49	48	simprbi 479	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:(1...(#‘𝐴))–1-1-onto→𝐴 → ^◡𝑔 Fn 𝐴)
50		fnresdm 5914	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → (^◡𝑔 ↾ 𝐴) = ^◡𝑔)
53		simpl3 1059	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → ¬ 𝑍 ∈ 𝐴)
54	14	cnveqi 5219	. . . . . . . . . . . . . . . . . 18 ⊢ ^◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ^◡((ℕ ∖ (1...(#‘𝐴))) × {𝑍})
55		cnvxp 5470	. . . . . . . . . . . . . . . . . 18 ⊢ ^◡((ℕ ∖ (1...(#‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(#‘𝐴))))
56	54, 55	eqtri 2632	. . . . . . . . . . . . . . . . 17 ⊢ ^◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(#‘𝐴))))
57	56	reseq1i 5313	. . . . . . . . . . . . . . . 16 ⊢ (^◡(𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(#‘𝐴)))) ↾ 𝐴)
58		incom 3767	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59		disjsn 4192	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍 ∈ 𝐴)
60	59	biimpri 217	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑍 ∈ 𝐴 → (𝐴 ∩ {𝑍}) = ∅)
61	58, 60	syl5eqr 2658	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑍 ∈ 𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62		xpdisjres 28793	. . . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . . 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 3730	. . . . . . . . . . . . 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 3919	. . . . . . . . . . . . 13 ⊢ (^◡𝑔 ∪ ∅) = ^◡𝑔
68	66, 67	syl6eq 2660	. . . . . . . . . . . 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 2656	. . . . . . . . . . 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 5825	. . . . . . . . . 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 246	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(#‘𝐴))–1-1-onto→𝐴) → Fun (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
72		vex 3176	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
73		nnex 10903	. . . . . . . . . . . . 13 ⊢ ℕ ∈ V
74		difexg 4735	. . . . . . . . . . . . 13 ⊢ (ℕ ∈ V → (ℕ ∖ (1...(#‘𝐴))) ∈ V)
75	73, 74	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℕ ∖ (1...(#‘𝐴))) ∈ V
76	75	mptex 6390	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍) ∈ V
77	72, 76	unex 6854	. . . . . . . . . 10 ⊢ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ∈ V
78		feq1 5939	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79		rneq 5272	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
80	79	sseq2d 3596	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓 ↔ 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍))))
81		cnveq 5218	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → ^◡𝑓 = ^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)))
82		eqidd 2611	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
83	81, 82	reseq12d 5318	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (^◡𝑓 ↾ 𝐴) = (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
84	83	funeqd 5825	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) → (Fun (^◡𝑓 ↾ 𝐴) ↔ Fun (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(#‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
85	78, 80, 84	3anbi123d 1391	. . . . . . . . . 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 3273	. . . . . . . . 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 1318	. . . . . . . 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 449	. . . . . . 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 2074	. . . . . 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 1259	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
92		nnenom 12641	. . . . . . . 8 ⊢ ℕ ≈ ω
93		simpl 472	. . . . . . . . 9 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → 𝐴 ≈ ω)
94	93	ensymd 7893	. . . . . . . 8 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ω ≈ 𝐴)
95		entr 7894	. . . . . . . 8 ⊢ ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
96	92, 94, 95	sylancr 694	. . . . . . 7 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ℕ ≈ 𝐴)
97		bren 7850	. . . . . . 7 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
98	96, 97	sylib 207	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
99		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑓(𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉)
100		simpr 476	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ–1-1-onto→𝐴)
101		f1of 6050	. . . . . . . . . 10 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ⟶𝐴)
102		ssun1 3738	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑍})
103		fss 5969	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶𝐴 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑍})) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
104	102, 103	mpan2 703	. . . . . . . . . 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 6057	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ–onto→𝐴)
107		forn 6031	. . . . . . . . . . 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 3617	. . . . . . . . 9 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝐴 ⊆ ran 𝑓)
110		f1ocnv 6062	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → ^◡𝑓:𝐴–1-1-onto→ℕ)
111		f1of1 6049	. . . . . . . . . . 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 6064	. . . . . . . . . . 11 ⊢ ((^◡𝑓:𝐴–1-1→ℕ ∧ 𝐴 ⊆ 𝐴) → (^◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(^◡𝑓 “ 𝐴))
114	29, 113	mpan2 703	. . . . . . . . . 10 ⊢ (^◡𝑓:𝐴–1-1→ℕ → (^◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(^◡𝑓 “ 𝐴))
115		f1ofun 6052	. . . . . . . . . 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 1235	. . . . . . . 8 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
118	117	ex 449	. . . . . . 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 2072	. . . . . 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 820	. . 3 ⊢ (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
123	122	3impia 1253	. 2 ⊢ (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
124	1, 123	syl3an1b 1354	1 ⊢ ((𝐴 ≼ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 ran crn 5039 ↾ cres 5040 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 Fincfn 7841 1c1 9816 ℕcn 10897 ...cfz 12197 #chash 12979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980

This theorem is referenced by: carsggect 29707

W3C validator