Theorem infpwfien 9984

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 9939	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2		infn0 9214	. . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≠ ∅)
4		fseqen 9949	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
5	1, 3, 4	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
6		xpdom1g 9014	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ (𝐴 × 𝐴))
7		domentr 8962	. . . . . . 7 ⊢ (((ω × 𝐴) ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≈ 𝐴) → (ω × 𝐴) ≼ 𝐴)
8	6, 1, 7	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ 𝐴)
9		endomtr 8961	. . . . . 6 ⊢ ((∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴)
10	5, 8, 9	syl2anc 585	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴)
11		numdom 9960	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom card)
12	10, 11	syldan 592	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom card)
13		eliun 4952	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛))
14		elmapi 8798	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴 ↑m 𝑛) → 𝑥:𝑛⟶𝐴)
15	14	ad2antll 730	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑥:𝑛⟶𝐴)
16	15	frnd 6678	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ⊆ 𝐴)
17		vex 3446	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
18	17	rnex 7862	. . . . . . . . . . . . . 14 ⊢ ran 𝑥 ∈ V
19	18	elpw 4560	. . . . . . . . . . . . 13 ⊢ (ran 𝑥 ∈ 𝒫 𝐴 ↔ ran 𝑥 ⊆ 𝐴)
20	16, 19	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ∈ 𝒫 𝐴)
21		simprl 771	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ ω)
22		ssid 3958	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ 𝑛
23		ssnnfi 9106	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑛) → 𝑛 ∈ Fin)
24	21, 22, 23	sylancl 587	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ Fin)
25		ffn 6670	. . . . . . . . . . . . . . 15 ⊢ (𝑥:𝑛⟶𝐴 → 𝑥 Fn 𝑛)
26		dffn4 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 Fn 𝑛 ↔ 𝑥:𝑛–onto→ran 𝑥)
27	25, 26	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝑥:𝑛⟶𝐴 → 𝑥:𝑛–onto→ran 𝑥)
28	15, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑥:𝑛–onto→ran 𝑥)
29		fofi 9225	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ Fin ∧ 𝑥:𝑛–onto→ran 𝑥) → ran 𝑥 ∈ Fin)
30	24, 28, 29	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ∈ Fin)
31	20, 30	elind 4154	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
32	31	expr 456	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑥 ∈ (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
33	32	rexlimdva 3139	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
34	13, 33	biimtrid 242	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
35	34	imp 406	. . . . . . 7 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
36	35	fmpttd 7069	. . . . . 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 6671	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) Fn ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
38	36	frnd 6678	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ⊆ (𝒫 𝐴 ∩ Fin))
39		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
40	39	elin2d 4159	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
41		isfi 8924	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Fin ↔ ∃𝑚 ∈ ω 𝑦 ≈ 𝑚)
42	40, 41	sylib 218	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑚 ∈ ω 𝑦 ≈ 𝑚)
43		ensym 8952	. . . . . . . . . . . . 13 ⊢ (𝑦 ≈ 𝑚 → 𝑚 ≈ 𝑦)
44		bren 8905	. . . . . . . . . . . . 13 ⊢ (𝑚 ≈ 𝑦 ↔ ∃𝑥 𝑥:𝑚–1-1-onto→𝑦)
45	43, 44	sylib 218	. . . . . . . . . . . 12 ⊢ (𝑦 ≈ 𝑚 → ∃𝑥 𝑥:𝑚–1-1-onto→𝑦)
46		simprl 771	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑚 ∈ ω)
47		f1of 6782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚⟶𝑦)
48	47	ad2antll 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝑦)
49		simplr 769	. . . . . . . . . . . . . . . . . . . . 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 4158	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ∈ 𝒫 𝐴)
51	50	elpwid 4565	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ⊆ 𝐴)
52	48, 51	fssd 6687	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝐴)
53		simplll 775	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝐴 ∈ dom card)
54		vex 3446	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑚 ∈ V
55		elmapg 8788	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ dom card ∧ 𝑚 ∈ V) → (𝑥 ∈ (𝐴 ↑m 𝑚) ↔ 𝑥:𝑚⟶𝐴))
56	53, 54, 55	sylancl 587	. . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥 ∈ (𝐴 ↑m 𝑚))
58		oveq2 7376	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝐴 ↑m 𝑛) = (𝐴 ↑m 𝑚))
59	58	eleq2d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 ↑m 𝑛) ↔ 𝑥 ∈ (𝐴 ↑m 𝑚)))
60	59	rspcev 3578	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑚)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛))
61	46, 57, 60	syl2anc 585	. . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . 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 6789	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚–onto→𝑦)
64	63	ad2antll 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚–onto→𝑦)
65		forn 6757	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥:𝑚–onto→𝑦 → ran 𝑥 = 𝑦)
66	64, 65	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → ran 𝑥 = 𝑦)
67	66	eqcomd 2743	. . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . 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 456	. . . . . . . . . . . . 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 1919	. . . . . . . . . . . 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 34	. . . . . . . . . . 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 3139	. . . . . . . . . 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 15	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))
74	73	ex 412	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)))
75		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥)
76	75	elrnmpt 5915	. . . . . . . . . 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 3447	. . . . . . . . 9 ⊢ (𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)𝑦 = ran 𝑥)
78		df-rex 3063	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)𝑦 = ran 𝑥 ↔ ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))
79	77, 78	bitri 275	. . . . . . . 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 252	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥)))
81	80	ssrdv 3941	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ⊆ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥))
82	38, 81	eqssd 3953	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))
83		df-fo 6506	. . . . 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 584	. . . 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 9979	. . . 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 65	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
87		domtr 8956	. . 3 ⊢ (((𝒫 𝐴 ∩ Fin) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
88	86, 10, 87	syl2anc 585	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
89		pwexg 5325	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝒫 𝐴 ∈ V)
90	89	adantr 480	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝒫 𝐴 ∈ V)
91		inex1g 5266	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
92	90, 91	syl 17	. . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
93		infpwfidom 9950	. . 3 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
94	92, 93	syl 17	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
95		sbth 9037	. 2 ⊢ (((𝒫 𝐴 ∩ Fin) ≼ 𝐴 ∧ 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
96	88, 94, 95	syl2anc 585	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  ∪ ciun 4948   class class class wbr 5100   ↦ cmpt 5181   × cxp 5630  dom cdm 5632  ran crn 5633   Fn wfn 6495  ⟶wf 6496  –onto→wfo 6498  –1-1-onto→wf1o 6499  (class class class)co 7368  ωcom 7818   ↑_m cmap 8775   ≈ cen 8892   ≼ cdom 8893  Fincfn 8895  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-seqom 8389  df-1o 8407  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-oi 9427  df-card 9863  df-acn 9866

This theorem is referenced by:  inffien  9985  isnumbasgrplem3  43459