padct - Mathbox for Thierry Arnoux

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Theorem padct 32635

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 9015	. 2 ⊢ (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2		nfv 1910	. . . . . 6 ⊢ Ⅎ𝑔(𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴)
3		nfv 1910	. . . . . 6 ⊢ Ⅎ𝑔∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))
4		isfinite2 9337	. . . . . . . . . 10 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin)
5		isfinite4 14381	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴)
6	4, 5	sylib 217	. . . . . . . . 9 ⊢ (𝐴 ≺ ω → (1...(♯‘𝐴)) ≈ 𝐴)
7	6	adantr 479	. . . . . . . 8 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (1...(♯‘𝐴)) ≈ 𝐴)
8		bren 8986	. . . . . . . 8 ⊢ ((1...(♯‘𝐴)) ≈ 𝐴 ↔ ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
9	7, 8	sylib 217	. . . . . . 7 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
10	9	3adant3 1129	. . . . . 6 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
11		f1of 6845	. . . . . . . . . . . 12 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(♯‘𝐴))⟶𝐴)
12	11	adantl 480	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(♯‘𝐴))⟶𝐴)
13		fconstmpt 5746	. . . . . . . . . . . . 13 ⊢ ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)
14	13	eqcomi 2735	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
15		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑍 ∈ 𝑉)
16		fconst2g 7222	. . . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(♯‘𝐴)))⟶{𝑍})
19		disjdif 4476	. . . . . . . . . . . 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 6766	. . . . . . . . . . 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 836	. . . . . . . . . 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 13588	. . . . . . . . . . . 12 ⊢ (1...(♯‘𝐴)) ⊆ ℕ
24		undif 4486	. . . . . . . . . . . 12 ⊢ ((1...(♯‘𝐴)) ⊆ ℕ ↔ ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ)
25	23, 24	mpbi 229	. . . . . . . . . . 11 ⊢ ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ
26	25	feq2i 6722	. . . . . . . . . 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 217	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
28	27	3adantl3 1165	. . . . . . . 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 4002	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ 𝐴
30		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
31		f1ofo 6852	. . . . . . . . . . . . . 14 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(♯‘𝐴))–onto→𝐴)
32		forn 6820	. . . . . . . . . . . . . 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 4033	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran 𝑔)
35	34	orcd 871	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝐴 ⊆ ran 𝑔 ∨ 𝐴 ⊆ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
36		ssun 4190	. . . . . . . . . . 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 6159	. . . . . . . . . 10 ⊢ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
39	37, 38	sseqtrrdi 4031	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
40	39	3adantl3 1165	. . . . . . . 8 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
41		dff1o3 6851	. . . . . . . . . . 11 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 ↔ (𝑔:(1...(♯‘𝐴))–onto→𝐴 ∧ Fun ^◡𝑔))
42	41	simprbi 495	. . . . . . . . . 10 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → Fun ^◡𝑔)
43	42	adantl 480	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → Fun ^◡𝑔)
44		cnvun 6156	. . . . . . . . . . . . 13 ⊢ ^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
45	44	reseq1i 5987	. . . . . . . . . . . 12 ⊢ (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46		resundir 6006	. . . . . . . . . . . 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 6853	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 ↔ (𝑔 Fn (1...(♯‘𝐴)) ∧ ^◡𝑔 Fn 𝐴))
49	48	simprbi 495	. . . . . . . . . . . . . . 15 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → ^◡𝑔 Fn 𝐴)
50		fnresdm 6682	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (^◡𝑔 ↾ 𝐴) = ^◡𝑔)
53		simpl3 1190	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → ¬ 𝑍 ∈ 𝐴)
54	14	cnveqi 5883	. . . . . . . . . . . . . . . . 17 ⊢ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ^◡((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
55		cnvxp 6170	. . . . . . . . . . . . . . . . 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 5987	. . . . . . . . . . . . . . 15 ⊢ (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴)
58		incom 4202	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59		disjsn 4720	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍 ∈ 𝐴)
60	59	biimpri 227	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑍 ∈ 𝐴 → (𝐴 ∩ {𝑍}) = ∅)
61	58, 60	eqtr3id 2780	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑍 ∈ 𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62		xpdisjres 32520	. . . . . . . . . . . . . . . 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 4164	. . . . . . . . . . . 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 4395	. . . . . . . . . . . 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 6583	. . . . . . . . 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 256	. . . . . . . 8 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → Fun (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
72		vex 3466	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
73		nnex 12272	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
74		difexg 5336	. . . . . . . . . . . 12 ⊢ (ℕ ∈ V → (ℕ ∖ (1...(♯‘𝐴))) ∈ V)
75	73, 74	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℕ ∖ (1...(♯‘𝐴))) ∈ V
76	75	mptex 7242	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ∈ V
77	72, 76	unex 7756	. . . . . . . . 9 ⊢ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ∈ V
78		feq1 6711	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79		rneq 5944	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
80	79	sseq2d 4012	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓 ↔ 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))))
81		cnveq 5882	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ^◡𝑓 = ^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
82		eqidd 2727	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
83	81, 82	reseq12d 5992	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (^◡𝑓 ↾ 𝐴) = (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
84	83	funeqd 6583	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (Fun (^◡𝑓 ↾ 𝐴) ↔ Fun (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)))
85	78, 80, 84	3anbi123d 1433	. . . . . . . . 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 3592	. . . . . . . 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 1368	. . . . . . 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 411	. . . . . 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 2208	. . . . 5 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
90	89	3expia 1118	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
91		nnenom 14002	. . . . . . . 8 ⊢ ℕ ≈ ω
92		simpl 481	. . . . . . . . 9 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → 𝐴 ≈ ω)
93	92	ensymd 9038	. . . . . . . 8 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ω ≈ 𝐴)
94		entr 9039	. . . . . . . 8 ⊢ ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
95	91, 93, 94	sylancr 585	. . . . . . 7 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ℕ ≈ 𝐴)
96		bren 8986	. . . . . . 7 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
97	95, 96	sylib 217	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
98		nfv 1910	. . . . . . 7 ⊢ Ⅎ𝑓(𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉)
99		simpr 483	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ–1-1-onto→𝐴)
100		f1of 6845	. . . . . . . . . 10 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ⟶𝐴)
101		ssun1 4173	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑍})
102		fss 6746	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶𝐴 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑍})) → 𝑓:ℕ⟶(𝐴 ∪ {𝑍}))
103	101, 102	mpan2 689	. . . . . . . . . 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 6852	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ–onto→𝐴)
106		forn 6820	. . . . . . . . . . 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 4033	. . . . . . . . 9 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝐴 ⊆ ran 𝑓)
109		f1ocnv 6857	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → ^◡𝑓:𝐴–1-1-onto→ℕ)
110		f1of1 6844	. . . . . . . . . . 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 6859	. . . . . . . . . . 11 ⊢ ((^◡𝑓:𝐴–1-1→ℕ ∧ 𝐴 ⊆ 𝐴) → (^◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(^◡𝑓 “ 𝐴))
113	29, 112	mpan2 689	. . . . . . . . . 10 ⊢ (^◡𝑓:𝐴–1-1→ℕ → (^◡𝑓 ↾ 𝐴):𝐴–1-1-onto→(^◡𝑓 “ 𝐴))
114		f1ofun 6847	. . . . . . . . . 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 1125	. . . . . . . 8 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
117	116	ex 411	. . . . . . 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 2205	. . . . . 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 954	. . 3 ⊢ (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
122	121	3impia 1114	. 2 ⊢ (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
123	1, 122	syl3an1b 1400	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 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3462 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4325 {csn 4633 class class class wbr 5155 ↦ cmpt 5238 × cxp 5682 ^◡ccnv 5683 ran crn 5685 ↾ cres 5686 “ cima 5687 Fun wfun 6550 Fn wfn 6551 ⟶wf 6552 –1-1→wf1 6553 –onto→wfo 6554 –1-1-onto→wf1o 6555 ‘cfv 6556 (class class class)co 7426 ωcom 7878 ≈ cen 8973 ≼ cdom 8974 ≺ csdm 8975 Fincfn 8976 1c1 11161 ℕcn 12266 ...cfz 13540 ♯chash 14349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-inf2 9686 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-n0 12527 df-z 12613 df-uz 12877 df-fz 13541 df-hash 14350

This theorem is referenced by: carsggect 34154

W3C validator