Theorem infpwfien 9998

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 9953	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2		infn0 9251	. . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
3	2	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≠ ∅)
4		fseqen 9963	. . . . . . 7 ⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
5	1, 3, 4	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
6		xpdom1g 9013	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ (𝐴 × 𝐴))
7		domentr 8953	. . . . . . 7 ⊢ (((ω × 𝐴) ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≈ 𝐴) → (ω × 𝐴) ≼ 𝐴)
8	6, 1, 7	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ 𝐴)
9		endomtr 8952	. . . . . 6 ⊢ ((∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴)
10	5, 8, 9	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴)
11		numdom 9974	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom card)
12	10, 11	syldan 591	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom card)
13		eliun 4958	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛))
14		elmapi 8787	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴 ↑m 𝑛) → 𝑥:𝑛⟶𝐴)
15	14	ad2antll 727	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑥:𝑛⟶𝐴)
16	15	frnd 6676	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ⊆ 𝐴)
17		vex 3449	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
18	17	rnex 7849	. . . . . . . . . . . . . 14 ⊢ ran 𝑥 ∈ V
19	18	elpw 4564	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ ω)
22		ssid 3966	. . . . . . . . . . . . . 14 ⊢ 𝑛 ⊆ 𝑛
23		ssnnfi 9113	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑛) → 𝑛 ∈ Fin)
24	21, 22, 23	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ Fin)
25		ffn 6668	. . . . . . . . . . . . . . 15 ⊢ (𝑥:𝑛⟶𝐴 → 𝑥 Fn 𝑛)
26		dffn4 6762	. . . . . . . . . . . . . . 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 9282	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ Fin ∧ 𝑥:𝑛–onto→ran 𝑥) → ran 𝑥 ∈ Fin)
30	24, 28, 29	syl2anc 584	. . . . . . . . . . . 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 457	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑥 ∈ (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
33	32	rexlimdva 3152	. . . . . . . . 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 407	. . . . . . 7 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
36	35	fmpttd 7063	. . . . . 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 6669	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) Fn ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
38	36	frnd 6676	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ⊆ (𝒫 𝐴 ∩ Fin))
39		simpr 485	. . . . . . . . . . . 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 8916	. . . . . . . . . . 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 8943	. . . . . . . . . . . . 13 ⊢ (𝑦 ≈ 𝑚 → 𝑚 ≈ 𝑦)
44		bren 8893	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑚 ∈ ω)
47		f1of 6784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚⟶𝑦)
48	47	ad2antll 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝑦)
49		simplr 767	. . . . . . . . . . . . . . . . . . . . 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 4569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ⊆ 𝐴)
52	48, 51	fssd 6686	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝐴)
53		simplll 773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝐴 ∈ dom card)
54		vex 3449	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑚 ∈ V
55		elmapg 8778	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ dom card ∧ 𝑚 ∈ V) → (𝑥 ∈ (𝐴 ↑m 𝑚) ↔ 𝑥:𝑚⟶𝐴))
56	53, 54, 55	sylancl 586	. . . . . . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑚)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛))
61	46, 57, 60	syl2anc 584	. . . . . . . . . . . . . . . 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 6791	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚–onto→𝑦)
64	63	ad2antll 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚–onto→𝑦)
65		forn 6759	. . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . 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 457	. . . . . . . . . . . . 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 1920	. . . . . . . . . . . 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 3152	. . . . . . . . . 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 413	. . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → ∃𝑥(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)))
75		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥)
76	75	elrnmpt 5911	. . . . . . . . . 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 3451	. . . . . . . . 9 ⊢ (𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)𝑦 = ran 𝑥)
78		df-rex 3074	. . . . . . . . 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	syl6ibr 251	. . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥)))
81	80	ssrdv 3950	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ⊆ ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥))
82	38, 81	eqssd 3961	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))
83		df-fo 6502	. . . . 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 583	. . . 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 9993	. . . 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 8947	. . 3 ⊢ (((𝒫 𝐴 ∩ Fin) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
88	86, 10, 87	syl2anc 584	. 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
89		pwexg 5333	. . . . 5 ⊢ (𝐴 ∈ dom card → 𝒫 𝐴 ∈ V)
90	89	adantr 481	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝒫 𝐴 ∈ V)
91		inex1g 5276	. . . 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 9964	. . 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 584	1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  ∪ ciun 4954   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634   Fn wfn 6491  ⟶wf 6492  –onto→wfo 6494  –1-1-onto→wf1o 6495  (class class class)co 7357  ωcom 7802   ↑_m cmap 8765   ≈ cen 8880   ≼ cdom 8881  Fincfn 8883  cardccrd 9871

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-seqom 8394  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-card 9875  df-acn 9878

This theorem is referenced by:  inffien  9999  isnumbasgrplem3  41418