Theorem pwfi2f1o 38367

Description: The pw2f1o 8276 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)

Hypotheses

Ref	Expression
pwfi2f1o.s	⊢ 𝑆 = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅}
pwfi2f1o.f	⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜}))

Assertion

Ref	Expression
pwfi2f1o	⊢ (𝐴 ∈ 𝑉 → 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑆   𝑥,𝑉,𝑦

Allowed substitution hints:   𝑆(𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem pwfi2f1o

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . 5 ⊢ (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))
2	1	pw2f1o2 38306	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3		f1of1 6323	. . . 4 ⊢ ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴 → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–1-1→𝒫 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–1-1→𝒫 𝐴)
5		pwfi2f1o.s	. . . 4 ⊢ 𝑆 = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅}
6		ssrab2 3849	. . . 4 ⊢ {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2𝑜 ↑𝑚 𝐴)
7	5, 6	eqsstri 3797	. . 3 ⊢ 𝑆 ⊆ (2𝑜 ↑𝑚 𝐴)
8		f1ores 6338	. . 3 ⊢ (((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–1-1→𝒫 𝐴 ∧ 𝑆 ⊆ (2𝑜 ↑𝑚 𝐴)) → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆))
9	4, 7, 8	sylancl 580	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆))
10		elmapfun 8088	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → Fun 𝑦)
11		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → 𝑦 ∈ (2𝑜 ↑𝑚 𝐴))
12		0ex 4952	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → ∅ ∈ V)
14	10, 11, 13	3jca 1158	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → (Fun 𝑦 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∧ ∅ ∈ V))
15	14	adantl 473	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (Fun 𝑦 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∧ ∅ ∈ V))
16		funisfsupp 8491	. . . . . . . . . . 11 ⊢ ((Fun 𝑦 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
18	13	anim2i 610	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝐴 ∈ 𝑉 ∧ ∅ ∈ V))
19		elmapi 8086	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → 𝑦:𝐴⟶2𝑜)
20	19	adantl 473	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → 𝑦:𝐴⟶2𝑜)
21		frnsuppeq 7513	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑦:𝐴⟶2𝑜 → (𝑦 supp ∅) = (◡𝑦 “ (2𝑜 ∖ {∅}))))
22	18, 20, 21	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 supp ∅) = (◡𝑦 “ (2𝑜 ∖ {∅})))
23		df-2o 7769	. . . . . . . . . . . . . . . 16 ⊢ 2𝑜 = suc 1𝑜
24		df-suc 5916	. . . . . . . . . . . . . . . . 17 ⊢ suc 1𝑜 = (1𝑜 ∪ {1𝑜})
25	24	equncomi 3923	. . . . . . . . . . . . . . . 16 ⊢ suc 1𝑜 = ({1𝑜} ∪ 1𝑜)
26	23, 25	eqtri 2787	. . . . . . . . . . . . . . 15 ⊢ 2𝑜 = ({1𝑜} ∪ 1𝑜)
27		df1o2 7781	. . . . . . . . . . . . . . . 16 ⊢ 1𝑜 = {∅}
28	27	eqcomi 2774	. . . . . . . . . . . . . . 15 ⊢ {∅} = 1𝑜
29	26, 28	difeq12i 3890	. . . . . . . . . . . . . 14 ⊢ (2𝑜 ∖ {∅}) = (({1𝑜} ∪ 1𝑜) ∖ 1𝑜)
30		difun2 4210	. . . . . . . . . . . . . . 15 ⊢ (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = ({1𝑜} ∖ 1𝑜)
31		incom 3969	. . . . . . . . . . . . . . . . 17 ⊢ ({1𝑜} ∩ 1𝑜) = (1𝑜 ∩ {1𝑜})
32		1on 7775	. . . . . . . . . . . . . . . . . . 19 ⊢ 1𝑜 ∈ On
33	32	onordi 6014	. . . . . . . . . . . . . . . . . 18 ⊢ Ord 1𝑜
34		orddisj 5948	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord 1𝑜 → (1𝑜 ∩ {1𝑜}) = ∅)
35	33, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1𝑜 ∩ {1𝑜}) = ∅
36	31, 35	eqtri 2787	. . . . . . . . . . . . . . . 16 ⊢ ({1𝑜} ∩ 1𝑜) = ∅
37		disj3 4184	. . . . . . . . . . . . . . . 16 ⊢ (({1𝑜} ∩ 1𝑜) = ∅ ↔ {1𝑜} = ({1𝑜} ∖ 1𝑜))
38	36, 37	mpbi 221	. . . . . . . . . . . . . . 15 ⊢ {1𝑜} = ({1𝑜} ∖ 1𝑜)
39	30, 38	eqtr4i 2790	. . . . . . . . . . . . . 14 ⊢ (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = {1𝑜}
40	29, 39	eqtri 2787	. . . . . . . . . . . . 13 ⊢ (2𝑜 ∖ {∅}) = {1𝑜}
41	40	imaeq2i 5648	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ (2𝑜 ∖ {∅})) = (◡𝑦 “ {1𝑜})
42	22, 41	syl6eq 2815	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 supp ∅) = (◡𝑦 “ {1𝑜}))
43	42	eleq1d 2829	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (◡𝑦 “ {1𝑜}) ∈ Fin))
44		cnvimass 5669	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ {1𝑜}) ⊆ dom 𝑦
45	44, 20	fssdm 6241	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (◡𝑦 “ {1𝑜}) ⊆ 𝐴)
46	45	biantrurd 528	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → ((◡𝑦 “ {1𝑜}) ∈ Fin ↔ ((◡𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (◡𝑦 “ {1𝑜}) ∈ Fin)))
47	17, 43, 46	3bitrd 296	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ ((◡𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (◡𝑦 “ {1𝑜}) ∈ Fin)))
48		elfpw 8479	. . . . . . . . 9 ⊢ ((◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((◡𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (◡𝑦 “ {1𝑜}) ∈ Fin))
49	47, 48	syl6bbr 280	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)))
50	49	rabbidva 3337	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ (◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)})
51		cnveq 5466	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
52	51	imaeq1d 5649	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (◡𝑥 “ {1𝑜}) = (◡𝑦 “ {1𝑜}))
53	52	cbvmptv 4911	. . . . . . . 8 ⊢ (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) = (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑦 “ {1𝑜}))
54	53	mptpreima 5816	. . . . . . 7 ⊢ (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ (◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)}
55	50, 5, 54	3eqtr4g 2824	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝑆 = (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)))
56	55	imaeq2d 5650	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆) = ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))))
57		f1ofo 6331	. . . . . . 7 ⊢ ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴 → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–onto→𝒫 𝐴)
58	2, 57	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–onto→𝒫 𝐴)
59		inss1 3994	. . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60		foimacnv 6341	. . . . . 6 ⊢ (((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–onto→𝒫 𝐴 ∧ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴) → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
61	58, 59, 60	sylancl 580	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
62	56, 61	eqtrd 2799	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63		f1oeq3 6316	. . . 4 ⊢ (((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆) = (𝒫 𝐴 ∩ Fin) → (((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆) ↔ ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
64	62, 63	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆) ↔ ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
65		resmpt 5628	. . . . . 6 ⊢ (𝑆 ⊆ (2𝑜 ↑𝑚 𝐴) → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜})))
66	7, 65	ax-mp 5	. . . . 5 ⊢ ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜}))
67		pwfi2f1o.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜}))
68	66, 67	eqtr4i 2790	. . . 4 ⊢ ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆) = 𝐹
69		f1oeq1 6314	. . . 4 ⊢ (((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆) = 𝐹 → (((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin) ↔ 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
70	68, 69	mp1i 13	. . 3 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin) ↔ 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
71	64, 70	bitrd 270	. 2 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆) ↔ 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
72	9, 71	mpbid 223	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  {crab 3059  Vcvv 3350   ∖ cdif 3731   ∪ cun 3732   ∩ cin 3733   ⊆ wss 3734  ∅c0 4081  𝒫 cpw 4317  {csn 4336   class class class wbr 4811   ↦ cmpt 4890  ^◡ccnv 5278   ↾ cres 5281   “ cima 5282  Ord word 5909  suc csuc 5912  Fun wfun 6064  ⟶wf 6066  –1-1→wf1 6067  –onto→wfo 6068  –1-1-onto→wf1o 6069  (class class class)co 6846   supp csupp 7501  1_𝑜c1o 7761  2_𝑜c2o 7762   ↑_𝑚 cmap 8064  Fincfn 8164   finSupp cfsupp 8486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ord 5913  df-on 5914  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-supp 7502  df-1o 7768  df-2o 7769  df-map 8066  df-fsupp 8487

This theorem is referenced by:  pwfi2en  38368