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Theorem padct 30774

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 8658	. 2 ⊢ (𝐴 ≼ ω ↔ (𝐴 ≺ ω ∨ 𝐴 ≈ ω))
2		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑔(𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴)
3		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑔∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))
4		isfinite2 8929	. . . . . . . . . 10 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin)
5		isfinite4 13929	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴)
6	4, 5	sylib 221	. . . . . . . . 9 ⊢ (𝐴 ≺ ω → (1...(♯‘𝐴)) ≈ 𝐴)
7	6	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (1...(♯‘𝐴)) ≈ 𝐴)
8		bren 8636	. . . . . . . 8 ⊢ ((1...(♯‘𝐴)) ≈ 𝐴 ↔ ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
9	7, 8	sylib 221	. . . . . . 7 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
10	9	3adant3 1134	. . . . . 6 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑔 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
11		f1of 6661	. . . . . . . . . . . 12 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(♯‘𝐴))⟶𝐴)
12	11	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(♯‘𝐴))⟶𝐴)
13		fconstmpt 5611	. . . . . . . . . . . . 13 ⊢ ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)
14	13	eqcomi 2746	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
15		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑍 ∈ 𝑉)
16		fconst2g 7018	. . . . . . . . . . . . 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 261	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍):(ℕ ∖ (1...(♯‘𝐴)))⟶{𝑍})
19		disjdif 4386	. . . . . . . . . . . 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 6581	. . . . . . . . . . 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 838	. . . . . . . . . 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 13143	. . . . . . . . . . . 12 ⊢ (1...(♯‘𝐴)) ⊆ ℕ
24		undif 4396	. . . . . . . . . . . 12 ⊢ ((1...(♯‘𝐴)) ⊆ ℕ ↔ ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ)
25	23, 24	mpbi 233	. . . . . . . . . . 11 ⊢ ((1...(♯‘𝐴)) ∪ (ℕ ∖ (1...(♯‘𝐴)))) = ℕ
26	25	feq2i 6537	. . . . . . . . . 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 221	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍}))
28	27	3adantl3 1170	. . . . . . . 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 3923	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ 𝐴
30		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴)
31		f1ofo 6668	. . . . . . . . . . . . . 14 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑔:(1...(♯‘𝐴))–onto→𝐴)
32		forn 6636	. . . . . . . . . . . . . 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 3954	. . . . . . . . . . . 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 4103	. . . . . . . . . . 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 6009	. . . . . . . . . 10 ⊢ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (ran 𝑔 ∪ ran (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
39	37, 38	sseqtrrdi 3952	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
40	39	3adantl3 1170	. . . . . . . 8 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
41		dff1o3 6667	. . . . . . . . . . 11 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 ↔ (𝑔:(1...(♯‘𝐴))–onto→𝐴 ∧ Fun ^◡𝑔))
42	41	simprbi 500	. . . . . . . . . 10 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → Fun ^◡𝑔)
43	42	adantl 485	. . . . . . . . 9 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → Fun ^◡𝑔)
44		cnvun 6006	. . . . . . . . . . . . 13 ⊢ ^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) = (^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))
45	44	reseq1i 5847	. . . . . . . . . . . 12 ⊢ (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴)
46		resundir 5866	. . . . . . . . . . . 12 ⊢ ((^◡𝑔 ∪ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ↾ 𝐴) ∪ (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
47	45, 46	eqtri 2765	. . . . . . . . . . 11 ⊢ (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴) = ((^◡𝑔 ↾ 𝐴) ∪ (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴))
48		dff1o4 6669	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 ↔ (𝑔 Fn (1...(♯‘𝐴)) ∧ ^◡𝑔 Fn 𝐴))
49	48	simprbi 500	. . . . . . . . . . . . . . 15 ⊢ (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴 → ^◡𝑔 Fn 𝐴)
50		fnresdm 6496	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → (^◡𝑔 ↾ 𝐴) = ^◡𝑔)
53		simpl3 1195	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → ¬ 𝑍 ∈ 𝐴)
54	14	cnveqi 5743	. . . . . . . . . . . . . . . . 17 ⊢ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ^◡((ℕ ∖ (1...(♯‘𝐴))) × {𝑍})
55		cnvxp 6020	. . . . . . . . . . . . . . . . 17 ⊢ ^◡((ℕ ∖ (1...(♯‘𝐴))) × {𝑍}) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
56	54, 55	eqtri 2765	. . . . . . . . . . . . . . . 16 ⊢ ^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) = ({𝑍} × (ℕ ∖ (1...(♯‘𝐴))))
57	56	reseq1i 5847	. . . . . . . . . . . . . . 15 ⊢ (^◡(𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ↾ 𝐴) = (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴)
58		incom 4115	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ {𝑍}) = ({𝑍} ∩ 𝐴)
59		disjsn 4627	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍 ∈ 𝐴)
60	59	biimpri 231	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑍 ∈ 𝐴 → (𝐴 ∩ {𝑍}) = ∅)
61	58, 60	eqtr3id 2792	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑍 ∈ 𝐴 → ({𝑍} ∩ 𝐴) = ∅)
62		xpdisjres 30656	. . . . . . . . . . . . . . . 16 ⊢ (({𝑍} ∩ 𝐴) = ∅ → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
63	61, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑍 ∈ 𝐴 → (({𝑍} × (ℕ ∖ (1...(♯‘𝐴)))) ↾ 𝐴) = ∅)
64	57, 63	syl5eq 2790	. . . . . . . . . . . . . 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 4078	. . . . . . . . . . . 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 4305	. . . . . . . . . . . 12 ⊢ (^◡𝑔 ∪ ∅) = ^◡𝑔
68	66, 67	eqtrdi 2794	. . . . . . . . . . 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	syl5eq 2790	. . . . . . . . . 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 6402	. . . . . . . . 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 260	. . . . . . . 8 ⊢ (((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐴) → Fun (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
72		vex 3412	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
73		nnex 11836	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
74		difexg 5220	. . . . . . . . . . . 12 ⊢ (ℕ ∈ V → (ℕ ∖ (1...(♯‘𝐴))) ∈ V)
75	73, 74	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℕ ∖ (1...(♯‘𝐴))) ∈ V
76	75	mptex 7039	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍) ∈ V
77	72, 76	unex 7531	. . . . . . . . 9 ⊢ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ∈ V
78		feq1 6526	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ↔ (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)):ℕ⟶(𝐴 ∪ {𝑍})))
79		rneq 5805	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ran 𝑓 = ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
80	79	sseq2d 3933	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (𝐴 ⊆ ran 𝑓 ↔ 𝐴 ⊆ ran (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍))))
81		cnveq 5742	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → ^◡𝑓 = ^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)))
82		eqidd 2738	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → 𝐴 = 𝐴)
83	81, 82	reseq12d 5852	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) → (^◡𝑓 ↾ 𝐴) = (^◡(𝑔 ∪ (𝑥 ∈ (ℕ ∖ (1...(♯‘𝐴))) ↦ 𝑍)) ↾ 𝐴))
84	83	funeqd 6402	. . . . . . . . . 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 3521	. . . . . . . 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 416	. . . . . 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 2217	. . . . 5 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
90	89	3expia 1123	. . . 4 ⊢ ((𝐴 ≺ ω ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
91		nnenom 13553	. . . . . . . 8 ⊢ ℕ ≈ ω
92		simpl 486	. . . . . . . . 9 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → 𝐴 ≈ ω)
93	92	ensymd 8679	. . . . . . . 8 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ω ≈ 𝐴)
94		entr 8680	. . . . . . . 8 ⊢ ((ℕ ≈ ω ∧ ω ≈ 𝐴) → ℕ ≈ 𝐴)
95	91, 93, 94	sylancr 590	. . . . . . 7 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ℕ ≈ 𝐴)
96		bren 8636	. . . . . . 7 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
97	95, 96	sylib 221	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) → ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
98		nfv 1922	. . . . . . 7 ⊢ Ⅎ𝑓(𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉)
99		simpr 488	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ–1-1-onto→𝐴)
100		f1of 6661	. . . . . . . . . 10 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ⟶𝐴)
101		ssun1 4086	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑍})
102		fss 6562	. . . . . . . . . . 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 6668	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ–onto→𝐴)
106		forn 6636	. . . . . . . . . . 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 3954	. . . . . . . . 9 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝐴 ⊆ ran 𝑓)
109		f1ocnv 6673	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → ^◡𝑓:𝐴–1-1-onto→ℕ)
110		f1of1 6660	. . . . . . . . . . 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 6675	. . . . . . . . . . 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 6663	. . . . . . . . . 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 1130	. . . . . . . 8 ⊢ (((𝐴 ≈ ω ∧ 𝑍 ∈ 𝑉) ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))
117	116	ex 416	. . . . . . 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 2214	. . . . . 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 957	. . 3 ⊢ (((𝐴 ≺ ω ∨ 𝐴 ≈ ω) ∧ 𝑍 ∈ 𝑉) → (¬ 𝑍 ∈ 𝐴 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴))))
122	121	3impia 1119	. 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 209 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2110 Vcvv 3408 ∖ cdif 3863 ∪ cun 3864 ∩ cin 3865 ⊆ wss 3866 ∅c0 4237 {csn 4541 class class class wbr 5053 ↦ cmpt 5135 × cxp 5549 ^◡ccnv 5550 ran crn 5552 ↾ cres 5553 “ cima 5554 Fun wfun 6374 Fn wfn 6375 ⟶wf 6376 –1-1→wf1 6377 –onto→wfo 6378 –1-1-onto→wf1o 6379 ‘cfv 6380 (class class class)co 7213 ωcom 7644 ≈ cen 8623 ≼ cdom 8624 ≺ csdm 8625 Fincfn 8626 1c1 10730 ℕcn 11830 ...cfz 13095 ♯chash 13896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-hash 13897

This theorem is referenced by: carsggect 31997

W3C validator