Theorem infpwfien 10042

Description: Any infinite well-orderable set is equinumerous to its set of finite subsets. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
infpwfien	⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)

Proof of Theorem infpwfien

Dummy variables 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infxpidm2 9997	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2		infn0 9258	. . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
3	2	adantl 486	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≠ ∅)
4		fseqen 10007	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
5	1, 3, 4	syl2anc 595	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
6		xpdom1g 9058	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ (𝐴 × 𝐴))
7		domentr 9006	. . . . . . 7 ⊢ (((ω × 𝐴) ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≈ 𝐴) → (ω × 𝐴) ≼ 𝐴)
8	6, 1, 7	syl2anc 595	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ 𝐴)
9		endomtr 9005	. . . . . 6 ⊢ ((∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴)
10	5, 8, 9	syl2anc 595	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴)
11		numdom 10018	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom card)
12	10, 11	syldan 602	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom card)
13		eliun 4961	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛))
14		elmapi 8842	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴 ↑m 𝑛) → 𝑥:𝑛⟶𝐴)
15	14	ad2antll 741	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑥:𝑛⟶𝐴)
16	15	frnd 6712	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ⊆ 𝐴)
17		vex 3467	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
18	17	rnex 7903	. . . . . . . . . . . . . 14 ⊢ ran 𝑥 ∈ V
19	18	elpw 4568	. . . . . . . . . . . . 13 ⊢ (ran 𝑥 ∈ 𝒫 𝐴 ↔ ran 𝑥 ⊆ 𝐴)
20	16, 19	sylibr 237	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ∈ 𝒫 𝐴)
21		simprl 782	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ ω)
22		ssid 3967	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ 𝑛
23		ssnnfi 9150	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑛) → 𝑛 ∈ Fin)
24	21, 22, 23	sylancl 597	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ Fin)
25		ffn 6703	. . . . . . . . . . . . . . 15 ⊢ (𝑥:𝑛⟶𝐴 → 𝑥 Fn 𝑛)
26		dffn4 6796	. . . . . . . . . . . . . . 15 ⊢ (𝑥 Fn 𝑛 ↔ 𝑥:𝑛–onto→ran 𝑥)
27	25, 26	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝑥:𝑛⟶𝐴 → 𝑥:𝑛–onto→ran 𝑥)
28	15, 27	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑥:𝑛–onto→ran 𝑥)
29		fofi 9269	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ Fin ∧ 𝑥:𝑛–onto→ran 𝑥) → ran 𝑥 ∈ Fin)
30	24, 28, 29	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ∈ Fin)
31	20, 30	elind 4161	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
32	31	expr 461	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑥 ∈ (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
33	32	rexlimdva 3172	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
34	13, 33	biimtrid 245	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
35	34	imp 411	. . . . . . 7 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
36	35	fmpttd 7108	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)⟶(𝒫 𝐴 ∩ Fin))
37	36	ffnd 6704	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) Fn ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
38	36	frnd 6712	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ⊆ (𝒫 𝐴 ∩ Fin))
39		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
40	39	elin2d 4166	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
41		isfi 8968	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Fin ↔ ∃𝑚 ∈ ω 𝑦 ≈ 𝑚)
42	40, 41	sylib 221	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑚 ∈ ω 𝑦 ≈ 𝑚)
43		ensym 8996	. . . . . . . . . . . . 13 ⊢ (𝑦 ≈ 𝑚 → 𝑚 ≈ 𝑦)
44		bren 8949	. . . . . . . . . . . . 13 ⊢ (𝑚 ≈ 𝑦 ↔ ∃𝑥 𝑥:𝑚–1-1-onto→𝑦)
45	43, 44	sylib 221	. . . . . . . . . . . 12 ⊢ (𝑦 ≈ 𝑚 → ∃𝑥 𝑥:𝑚–1-1-onto→𝑦)
46		simprl 782	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑚 ∈ ω)
47		f1of 6818	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚⟶𝑦)
48	47	ad2antll 741	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝑦)
49		simplr 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
50	49	elin1d 4165	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ∈ 𝒫 𝐴)
51	50	elpwid 4573	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ⊆ 𝐴)
52	48, 51	fssd 6721	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝐴)
53		simplll 786	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝐴 ∈ dom card)
54		vex 3467	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑚 ∈ V
55		elmapg 8832	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ dom card ∧ 𝑚 ∈ V) → (𝑥 ∈ (𝐴 ↑m 𝑚) ↔ 𝑥:𝑚⟶𝐴))
56	53, 54, 55	sylancl 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → (𝑥 ∈ (𝐴 ↑m 𝑚) ↔ 𝑥:𝑚⟶𝐴))
57	52, 56	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥 ∈ (𝐴 ↑m 𝑚))
58		oveq2 7416	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝐴 ↑m 𝑛) = (𝐴 ↑m 𝑚))
59	58	eleq2d 2855	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 ↑m 𝑛) ↔ 𝑥 ∈ (𝐴 ↑m 𝑚)))
60	59	rspcev 3590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑚)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛))
61	46, 57, 60	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛))
62	61, 13	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
63		f1ofo 6826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚–onto→𝑦)
64	63	ad2antll 741	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚–onto→𝑦)
65		forn 6793	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥:𝑚–onto→𝑦 → ran 𝑥 = 𝑦)
66	64, 65	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → ran 𝑥 = 𝑦)
67	66	eqcomd 2775	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 = ran 𝑥)
68	62, 67	jca 520	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))
69	68	expr 461	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (𝑥:𝑚–1-1-onto→𝑦 → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)))
70	69	eximdv 1944	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (∃𝑥 𝑥:𝑚–1-1-onto→𝑦 → ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)))
71	45, 70	syl5 35	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (𝑦 ≈ 𝑚 → ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)))
72	71	rexlimdva 3172	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (∃𝑚 ∈ ω 𝑦 ≈ 𝑚 → ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)))
73	42, 72	mpd 16	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))
74	73	ex 417	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)))
75		eqid 2769	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥)
76	75	elrnmpt 5946	. . . . . . . . . 10 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)𝑦 = ran 𝑥))
77	76	elv 3468	. . . . . . . . 9 ⊢ (𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)𝑦 = ran 𝑥)
78		df-rex 3096	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)𝑦 = ran 𝑥 ↔ ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))
79	77, 78	bitri 278	. . . . . . . 8 ⊢ (𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))
80	74, 79	imbitrrdi 255	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥)))
81	80	ssrdv 3951	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ⊆ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥))
82	38, 81	eqssd 3962	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))
83		df-fo 6539	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)–onto→(𝒫 𝐴 ∩ Fin) ↔ ((𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) Fn ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin)))
84	37, 82, 83	sylanbrc 594	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)–onto→(𝒫 𝐴 ∩ Fin))
85		fodomnum 10037	. . . 4 ⊢ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom card → ((𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)–onto→(𝒫 𝐴 ∩ Fin) → (𝒫 𝐴 ∩ Fin) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
86	12, 84, 85	sylc 66	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
87		domtr 9000	. . 3 ⊢ (((𝒫 𝐴 ∩ Fin) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
88	86, 10, 87	syl2anc 595	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
89		pwexg 5347	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝒫 𝐴 ∈ V)
90	89	adantr 485	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝒫 𝐴 ∈ V)
91		inex1g 5287	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
92	90, 91	syl 18	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
93		infpwfidom 10008	. . 3 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
94	92, 93	syl 18	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
95		sbth 9081	. 2 ⊢ (((𝒫 𝐴 ∩ Fin) ≼ 𝐴 ∧ 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
96	88, 94, 95	syl2anc 595	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4564  ∪ ciun 4957   class class class wbr 5110   ↦ cmpt 5193   × cxp 5657  dom cdm 5659  ran crn 5660   Fn wfn 6528  ⟶wf 6529  –onto→wfo 6531  –1-1-onto→wf1o 6532  (class class class)co 7408  ωcom 7858   ↑_m cmap 8820   ≈ cen 8936   ≼ cdom 8937  Fincfn 8939  cardccrd 9917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-seqom 8431  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9468  df-card 9921  df-acn 9924

This theorem is referenced by:  inffien  10043  isnumbasgrplem3  43717