infpwfien - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > infpwfien		Structured version Visualization version GIF version

Theorem infpwfien 10059

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 10014	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2		infn0 9309	. . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
3	2	adantl 480	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≠ ∅)
4		fseqen 10024	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ≈ (ω × 𝐴))
5	1, 3, 4	syl2anc 582	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ≈ (ω × 𝐴))
6		xpdom1g 9071	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ (𝐴 × 𝐴))
7		domentr 9011	. . . . . . 7 ⊢ (((ω × 𝐴) ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≈ 𝐴) → (ω × 𝐴) ≼ 𝐴)
8	6, 1, 7	syl2anc 582	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ 𝐴)
9		endomtr 9010	. . . . . 6 ⊢ ((∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ≈ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ≼ 𝐴)
10	5, 8, 9	syl2anc 582	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ≼ 𝐴)
11		numdom 10035	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ dom card)
12	10, 11	syldan 589	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ dom card)
13		eliun 5000	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑_m 𝑛))
14		elmapi 8845	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴 ↑_m 𝑛) → 𝑥:𝑛⟶𝐴)
15	14	ad2antll 725	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑_m 𝑛))) → 𝑥:𝑛⟶𝐴)
16	15	frnd 6724	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑_m 𝑛))) → ran 𝑥 ⊆ 𝐴)
17		vex 3476	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
18	17	rnex 7905	. . . . . . . . . . . . . 14 ⊢ ran 𝑥 ∈ V
19	18	elpw 4605	. . . . . . . . . . . . 13 ⊢ (ran 𝑥 ∈ 𝒫 𝐴 ↔ ran 𝑥 ⊆ 𝐴)
20	16, 19	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑_m 𝑛))) → ran 𝑥 ∈ 𝒫 𝐴)
21		simprl 767	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑_m 𝑛))) → 𝑛 ∈ ω)
22		ssid 4003	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ 𝑛
23		ssnnfi 9171	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑛) → 𝑛 ∈ Fin)
24	21, 22, 23	sylancl 584	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑_m 𝑛))) → 𝑛 ∈ Fin)
25		ffn 6716	. . . . . . . . . . . . . . 15 ⊢ (𝑥:𝑛⟶𝐴 → 𝑥 Fn 𝑛)
26		dffn4 6810	. . . . . . . . . . . . . . 15 ⊢ (𝑥 Fn 𝑛 ↔ 𝑥:𝑛–onto→ran 𝑥)
27	25, 26	sylib 217	. . . . . . . . . . . . . 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 9340	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ Fin ∧ 𝑥:𝑛–onto→ran 𝑥) → ran 𝑥 ∈ Fin)
30	24, 28, 29	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑_m 𝑛))) → ran 𝑥 ∈ Fin)
31	20, 30	elind 4193	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑_m 𝑛))) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
32	31	expr 455	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑥 ∈ (𝐴 ↑_m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
33	32	rexlimdva 3153	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑_m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
34	13, 33	biimtrid 241	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
35	34	imp 405	. . . . . . 7 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
36	35	fmpttd 7115	. . . . . 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 6717	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↦ ran 𝑥) Fn ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
38	36	frnd 6724	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↦ ran 𝑥) ⊆ (𝒫 𝐴 ∩ Fin))
39		simpr 483	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
40	39	elin2d 4198	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
41		isfi 8974	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Fin ↔ ∃𝑚 ∈ ω 𝑦 ≈ 𝑚)
42	40, 41	sylib 217	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑚 ∈ ω 𝑦 ≈ 𝑚)
43		ensym 9001	. . . . . . . . . . . . 13 ⊢ (𝑦 ≈ 𝑚 → 𝑚 ≈ 𝑦)
44		bren 8951	. . . . . . . . . . . . 13 ⊢ (𝑚 ≈ 𝑦 ↔ ∃𝑥 𝑥:𝑚–1-1-onto→𝑦)
45	43, 44	sylib 217	. . . . . . . . . . . 12 ⊢ (𝑦 ≈ 𝑚 → ∃𝑥 𝑥:𝑚–1-1-onto→𝑦)
46		simprl 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑚 ∈ ω)
47		f1of 6832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚⟶𝑦)
48	47	ad2antll 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝑦)
49		simplr 765	. . . . . . . . . . . . . . . . . . . . 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 4197	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ∈ 𝒫 𝐴)
51	50	elpwid 4610	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ⊆ 𝐴)
52	48, 51	fssd 6734	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝐴)
53		simplll 771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝐴 ∈ dom card)
54		vex 3476	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑚 ∈ V
55		elmapg 8835	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ dom card ∧ 𝑚 ∈ V) → (𝑥 ∈ (𝐴 ↑_m 𝑚) ↔ 𝑥:𝑚⟶𝐴))
56	53, 54, 55	sylancl 584	. . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥 ∈ (𝐴 ↑_m 𝑚))
58		oveq2 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝐴 ↑_m 𝑛) = (𝐴 ↑_m 𝑚))
59	58	eleq2d 2817	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 ↑_m 𝑛) ↔ 𝑥 ∈ (𝐴 ↑_m 𝑚)))
60	59	rspcev 3611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑_m 𝑚)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑_m 𝑛))
61	46, 57, 60	syl2anc 582	. . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚–onto→𝑦)
64	63	ad2antll 725	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚–onto→𝑦)
65		forn 6807	. . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 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 510	. . . . . . . . . . . . . 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 455	. . . . . . . . . . . . 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 1918	. . . . . . . . . . . 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 3153	. . . . . . . . . 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 411	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∧ 𝑦 = ran 𝑥)))
75		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↦ ran 𝑥) = (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↦ ran 𝑥)
76	75	elrnmpt 5954	. . . . . . . . . 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 3478	. . . . . . . . 9 ⊢ (𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)𝑦 = ran 𝑥)
78		df-rex 3069	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)𝑦 = ran 𝑥 ↔ ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∧ 𝑦 = ran 𝑥))
79	77, 78	bitri 274	. . . . . . . 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 251	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↦ ran 𝑥)))
81	80	ssrdv 3987	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ⊆ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↦ ran 𝑥))
82	38, 81	eqssd 3998	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))
83		df-fo 6548	. . . . 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 581	. . . 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 10054	. . . 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 9005	. . 3 ⊢ (((𝒫 𝐴 ∩ Fin) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
88	86, 10, 87	syl2anc 582	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
89		pwexg 5375	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝒫 𝐴 ∈ V)
90	89	adantr 479	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝒫 𝐴 ∈ V)
91		inex1g 5318	. . . 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 10025	. . 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 9095	. 2 ⊢ (((𝒫 𝐴 ∩ Fin) ≼ 𝐴 ∧ 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
96	88, 94, 95	syl2anc 582	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ≠ wne 2938 ∃wrex 3068 Vcvv 3472 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 ∪ ciun 4996 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 dom cdm 5675 ran crn 5676 Fn wfn 6537 ⟶wf 6538 –onto→wfo 6540 –1-1-onto→wf1o 6541 (class class class)co 7411 ωcom 7857 ↑_m cmap 8822 ≈ cen 8938 ≼ cdom 8939 Fincfn 8941 cardccrd 9932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-seqom 8450 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-oi 9507 df-card 9936 df-acn 9939

This theorem is referenced by: inffien 10060 isnumbasgrplem3 42149

W3C validator