padct - Mathbox for Thierry Arnoux

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Theorem padct 32701

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 8904	. 2 ⊢ (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑔(𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴)
3		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑔∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))
4		isfinite2 9182	. . . . . . . . . 10 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin)
5		isfinite4 14269	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴)
6	4, 5	sylib 218	. . . . . . . . 9 ⊢ (𝐴 ≺ ω → (1...(♯‘𝐴)) ≈ 𝐴)
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (1...(♯‘𝐴)) ≈ 𝐴)
8		bren 8879	. . . . . . . 8 ⊢ ((1...(♯‘𝐴)) ≈ 𝐴 ↔ ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
9	7, 8	sylib 218	. . . . . . 7 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
10	9	3adant3 1132	. . . . . 6 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
11		f1of 6763	. . . . . . . . . . . 12 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(♯‘𝐴))⟶𝐴)
12	11	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(♯‘𝐴))⟶𝐴)
13		fconstmpt 5676	. . . . . . . . . . . . 13 ⊢ ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)
14	13	eqcomi 2740	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
15		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑍 ∈ 𝑉)
16		fconst2g 7137	. . . . . . . . . . . . 13 ⊢ (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	. . . . . . . . . . . 12 ⊢ (((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 258	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(♯‘𝐴)))⟶{𝑍})
19		disjdif 4419	. . . . . . . . . . . 12 ⊢ ((1...(♯‘𝐴)) ∩ (ℕ ∖ (1...(♯‘𝐴)))) = ∅
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → ((1...(♯‘𝐴)) ∩ (ℕ ∖ (1...(♯‘𝐴)))) = ∅)
21		fun 6685	. . . . . . . . . . 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}))
22	12, 18, 20, 21	syl21anc 837	. . . . . . . . . 10 ⊢ (((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 13455	. . . . . . . . . . . 12 ⊢ (1...(♯‘𝐴)) ⊆ ℕ
24		undif 4429	. . . . . . . . . . . 12 ⊢ ((1...(♯‘𝐴)) ⊆ ℕ ↔ ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ)
25	23, 24	mpbi 230	. . . . . . . . . . 11 ⊢ ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ
26	25	feq2i 6643	. . . . . . . . . 10 ⊢ ((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 218	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
28	27	3adantl3 1169	. . . . . . . 8 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
29		ssid 3952	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ 𝐴
30		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
31		f1ofo 6770	. . . . . . . . . . . . . 14 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(♯‘𝐴))–onto→𝐴)
32		forn 6738	. . . . . . . . . . . . . 14 ⊢ (𝑔:(1...(♯‘𝐴))–onto→𝐴 → ran 𝑔 = 𝐴)
33	30, 31, 32	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → ran 𝑔 = 𝐴)
34	29, 33	sseqtrrid 3973	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran 𝑔)
35	34	orcd 873	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝐴 ⊆ ran 𝑔 ∨ 𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
36		ssun 4142	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ran 𝑔 ∨ 𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝐴 ⊆ (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
37	35, 36	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
38		rnun 6092	. . . . . . . . . 10 ⊢ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
39	37, 38	sseqtrrdi 3971	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
40	39	3adantl3 1169	. . . . . . . 8 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
41		dff1o3 6769	. . . . . . . . . . 11 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 ↔ (𝑔:(1...(♯‘𝐴))–onto→𝐴 ∧ Fun ^◡𝑔))
42	41	simprbi 496	. . . . . . . . . 10 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → Fun ^◡𝑔)
43	42	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → Fun ^◡𝑔)
44		cnvun 6089	. . . . . . . . . . . . 13 ⊢ ^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
45	44	reseq1i 5923	. . . . . . . . . . . 12 ⊢ (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46		resundir 5942	. . . . . . . . . . . 12 ⊢ ((^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ↾ 𝐴) ∪ (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
47	45, 46	eqtri 2754	. . . . . . . . . . 11 ⊢ (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ↾ 𝐴) ∪ (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
48		dff1o4 6771	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 ↔ (𝑔 Fn (1...(♯‘𝐴)) ∧ ^◡𝑔 Fn 𝐴))
49	48	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → ^◡𝑔 Fn 𝐴)
50		fnresdm 6600	. . . . . . . . . . . . . . 15 ⊢ (^◡𝑔 Fn 𝐴 → (^◡𝑔 ↾ 𝐴) = ^◡𝑔)
51	49, 50	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → (^◡𝑔 ↾ 𝐴) = ^◡𝑔)
52	51	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (^◡𝑔 ↾ 𝐴) = ^◡𝑔)
53		simpl3 1194	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → ¬ 𝑍 ∈ 𝐴)
54	14	cnveqi 5813	. . . . . . . . . . . . . . . . 17 ⊢ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ^◡((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
55		cnvxp 6104	. . . . . . . . . . . . . . . . 17 ⊢ ^◡((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
56	54, 55	eqtri 2754	. . . . . . . . . . . . . . . 16 ⊢ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
57	56	reseq1i 5923	. . . . . . . . . . . . . . 15 ⊢ (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴)
58		incom 4156	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59		disjsn 4661	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍 ∈ 𝐴)
60	59	biimpri 228	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑍 ∈ 𝐴 → (𝐴 ∩ {𝑍}) = ∅)
61	58, 60	eqtr3id 2780	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑍 ∈ 𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62		xpdisjres 32578	. . . . . . . . . . . . . . . 16 ⊢ (({𝑍} ∩ 𝐴) = ∅ → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
63	61, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑍 ∈ 𝐴 → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
64	57, 63	eqtrid 2778	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑍 ∈ 𝐴 → (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = ∅)
65	53, 64	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = ∅)
66	52, 65	uneq12d 4116	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → ((^◡𝑔 ↾ 𝐴) ∪ (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴)) = (^◡𝑔 ∪ ∅))
67		un0 4341	. . . . . . . . . . . 12 ⊢ (^◡𝑔 ∪ ∅) = ^◡𝑔
68	66, 67	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (((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	eqtrid 2778	. . . . . . . . . 10 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ^◡𝑔)
70	69	funeqd 6503	. . . . . . . . 9 ⊢ (((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 257	. . . . . . . 8 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → Fun (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
72		vex 3440	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
73		nnex 12131	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
74		difexg 5265	. . . . . . . . . . . 12 ⊢ (ℕ ∈ V → (ℕ ∖ (1...(♯‘𝐴))) ∈ V)
75	73, 74	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℕ ∖ (1...(♯‘𝐴))) ∈ V
76	75	mptex 7157	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ∈ V
77	72, 76	unex 7677	. . . . . . . . 9 ⊢ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ∈ V
78		feq1 6629	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79		rneq 5875	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
80	79	sseq2d 3962	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓 ↔ 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))))
81		cnveq 5812	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ^◡𝑓 = ^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
82		eqidd 2732	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
83	81, 82	reseq12d 5928	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (^◡𝑓 ↾ 𝐴) = (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
84	83	funeqd 6503	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (Fun (^◡𝑓 ↾ 𝐴) ↔ Fun (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
85	78, 80, 84	3anbi123d 1438	. . . . . . . . 9 ⊢ (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 3556	. . . . . . . 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 (^◡𝑓 ↾ 𝐴)))
87	28, 40, 71, 86	syl3anc 1373	. . . . . . 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)))
88	87	ex 412	. . . . . 6 ⊢ ((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	2, 3, 10, 88	exlimimdd 2222	. . . . 5 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
90	89	3expia 1121	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
91		nnenom 13887	. . . . . . . 8 ⊢ ℕ ≈ ω
92		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → 𝐴 ≈ ω)
93	92	ensymd 8927	. . . . . . . 8 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ω ≈ 𝐴)
94		entr 8928	. . . . . . . 8 ⊢ ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
95	91, 93, 94	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ℕ ≈ 𝐴)
96		bren 8879	. . . . . . 7 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
97	95, 96	sylib 218	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
98		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑓(𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉)
99		simpr 484	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ–1-1-onto→𝐴)
100		f1of 6763	. . . . . . . . . 10 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ⟶𝐴)
101		ssun1 4125	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑍})
102		fss 6667	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶𝐴 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑍})) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
103	101, 102	mpan2 691	. . . . . . . . . 10 ⊢ (𝑓:ℕ⟶𝐴 → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
104	99, 100, 103	3syl 18	. . . . . . . . 9 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
105		f1ofo 6770	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ–onto→𝐴)
106		forn 6738	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → ran 𝑓 = 𝐴)
107	99, 105, 106	3syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran 𝑓 = 𝐴)
108	29, 107	sseqtrrid 3973	. . . . . . . . 9 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝐴 ⊆ ran 𝑓)
109		f1ocnv 6775	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → ^◡𝑓:𝐴–1-1-onto→ℕ)
110		f1of1 6762	. . . . . . . . . . 11 ⊢ (^◡𝑓:𝐴–1-1-onto→ℕ → ^◡𝑓:𝐴–1-1→ℕ)
111	99, 109, 110	3syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → ^◡𝑓:𝐴–1-1→ℕ)
112		f1ores 6777	. . . . . . . . . . 11 ⊢ ((^◡𝑓:𝐴–1-1→ℕ ∧ 𝐴 ⊆ 𝐴) → (^◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(^◡𝑓 “ 𝐴))
113	29, 112	mpan2 691	. . . . . . . . . 10 ⊢ (^◡𝑓:𝐴–1-1→ℕ → (^◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(^◡𝑓 “ 𝐴))
114		f1ofun 6765	. . . . . . . . . 10 ⊢ ((^◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(^◡𝑓 “ 𝐴) → Fun (^◡𝑓 ↾ 𝐴))
115	111, 113, 114	3syl 18	. . . . . . . . 9 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → Fun (^◡𝑓 ↾ 𝐴))
116	104, 108, 115	3jca 1128	. . . . . . . 8 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
117	116	ex 412	. . . . . . 7 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → (𝑓:ℕ–1-1-onto→𝐴 → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
118	98, 117	eximd 2219	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → (∃𝑓 𝑓:ℕ–1-1-onto→𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
119	97, 118	mpd 15	. . . . 5 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
120	119	a1d 25	. . . 4 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
121	90, 120	jaoian 958	. . 3 ⊢ (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
122	121	3impia 1117	. 2 ⊢ (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
123	1, 122	syl3an1b 1405	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2111 Vcvv 3436 ∖ cdif 3894 ∪ cun 3895 ∩ cin 3896 ⊆ wss 3897 ∅c0 4280 {csn 4573 class class class wbr 5089 ↦ cmpt 5170 × cxp 5612 ^◡ccnv 5613 ran crn 5615 ↾ cres 5616 “ cima 5617 Fun wfun 6475 Fn wfn 6476 ⟶wf 6477 –1-1→wf1 6478 –onto→wfo 6479 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 ωcom 7796 ≈ cen 8866 ≼ cdom 8867 ≺ csdm 8868 Fincfn 8869 1c1 11007 ℕcn 12125 ...cfz 13407 ♯chash 14237

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-hash 14238

This theorem is referenced by: carsggect 34331

W3C validator