Theorem pwfi2f1o 37666

Description: The pw2f1o 8065 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 2622	. . . . 5 ⊢ (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))
2	1	pw2f1o2 37605	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3		f1of1 6136	. . . 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 3687	. . . 4 ⊢ {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2𝑜 ↑𝑚 𝐴)
7	5, 6	eqsstri 3635	. . 3 ⊢ 𝑆 ⊆ (2𝑜 ↑𝑚 𝐴)
8		f1ores 6151	. . 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 694	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆))
10		elmapfun 7881	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → Fun 𝑦)
11		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → 𝑦 ∈ (2𝑜 ↑𝑚 𝐴))
12		0ex 4790	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → ∅ ∈ V)
14	10, 11, 13	3jca 1242	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → (Fun 𝑦 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∧ ∅ ∈ V))
15	14	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (Fun 𝑦 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∧ ∅ ∈ V))
16		funisfsupp 8280	. . . . . . . . . . 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 593	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝐴 ∈ 𝑉 ∧ ∅ ∈ V))
19		elmapi 7879	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → 𝑦:𝐴⟶2𝑜)
20	19	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → 𝑦:𝐴⟶2𝑜)
21		frnsuppeq 7307	. . . . . . . . . . . . 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 7561	. . . . . . . . . . . . . . . 16 ⊢ 2𝑜 = suc 1𝑜
24		df-suc 5729	. . . . . . . . . . . . . . . . 17 ⊢ suc 1𝑜 = (1𝑜 ∪ {1𝑜})
25	24	equncomi 3759	. . . . . . . . . . . . . . . 16 ⊢ suc 1𝑜 = ({1𝑜} ∪ 1𝑜)
26	23, 25	eqtri 2644	. . . . . . . . . . . . . . 15 ⊢ 2𝑜 = ({1𝑜} ∪ 1𝑜)
27		df1o2 7572	. . . . . . . . . . . . . . . 16 ⊢ 1𝑜 = {∅}
28	27	eqcomi 2631	. . . . . . . . . . . . . . 15 ⊢ {∅} = 1𝑜
29	26, 28	difeq12i 3726	. . . . . . . . . . . . . 14 ⊢ (2𝑜 ∖ {∅}) = (({1𝑜} ∪ 1𝑜) ∖ 1𝑜)
30		difun2 4048	. . . . . . . . . . . . . . 15 ⊢ (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = ({1𝑜} ∖ 1𝑜)
31		incom 3805	. . . . . . . . . . . . . . . . 17 ⊢ ({1𝑜} ∩ 1𝑜) = (1𝑜 ∩ {1𝑜})
32		1on 7567	. . . . . . . . . . . . . . . . . . 19 ⊢ 1𝑜 ∈ On
33	32	onordi 5832	. . . . . . . . . . . . . . . . . 18 ⊢ Ord 1𝑜
34		orddisj 5762	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord 1𝑜 → (1𝑜 ∩ {1𝑜}) = ∅)
35	33, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1𝑜 ∩ {1𝑜}) = ∅
36	31, 35	eqtri 2644	. . . . . . . . . . . . . . . 16 ⊢ ({1𝑜} ∩ 1𝑜) = ∅
37		disj3 4021	. . . . . . . . . . . . . . . 16 ⊢ (({1𝑜} ∩ 1𝑜) = ∅ ↔ {1𝑜} = ({1𝑜} ∖ 1𝑜))
38	36, 37	mpbi 220	. . . . . . . . . . . . . . 15 ⊢ {1𝑜} = ({1𝑜} ∖ 1𝑜)
39	30, 38	eqtr4i 2647	. . . . . . . . . . . . . 14 ⊢ (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = {1𝑜}
40	29, 39	eqtri 2644	. . . . . . . . . . . . 13 ⊢ (2𝑜 ∖ {∅}) = {1𝑜}
41	40	imaeq2i 5464	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ (2𝑜 ∖ {∅})) = (◡𝑦 “ {1𝑜})
42	22, 41	syl6eq 2672	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 supp ∅) = (◡𝑦 “ {1𝑜}))
43	42	eleq1d 2686	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (◡𝑦 “ {1𝑜}) ∈ Fin))
44		cnvimass 5485	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ {1𝑜}) ⊆ dom 𝑦
45		fdm 6051	. . . . . . . . . . . . 13 ⊢ (𝑦:𝐴⟶2𝑜 → dom 𝑦 = 𝐴)
46	20, 45	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → dom 𝑦 = 𝐴)
47	44, 46	syl5sseq 3653	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (◡𝑦 “ {1𝑜}) ⊆ 𝐴)
48	47	biantrurd 529	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → ((◡𝑦 “ {1𝑜}) ∈ Fin ↔ ((◡𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (◡𝑦 “ {1𝑜}) ∈ Fin)))
49	17, 43, 48	3bitrd 294	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ ((◡𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (◡𝑦 “ {1𝑜}) ∈ Fin)))
50		elfpw 8268	. . . . . . . . 9 ⊢ ((◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((◡𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (◡𝑦 “ {1𝑜}) ∈ Fin))
51	49, 50	syl6bbr 278	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)))
52	51	rabbidva 3188	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ (◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)})
53		cnveq 5296	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
54	53	imaeq1d 5465	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (◡𝑥 “ {1𝑜}) = (◡𝑦 “ {1𝑜}))
55	54	cbvmptv 4750	. . . . . . . 8 ⊢ (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) = (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑦 “ {1𝑜}))
56	55	mptpreima 5628	. . . . . . 7 ⊢ (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ (◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)}
57	52, 5, 56	3eqtr4g 2681	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝑆 = (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)))
58	57	imaeq2d 5466	. . . . 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))))
59		f1ofo 6144	. . . . . . 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)
60	2, 59	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–onto→𝒫 𝐴)
61		inss1 3833	. . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
62		foimacnv 6154	. . . . . 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))
63	60, 61, 62	sylancl 694	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
64	58, 63	eqtrd 2656	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
65		f1oeq3 6129	. . . 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)))
66	64, 65	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)))
67		resmpt 5449	. . . . . 6 ⊢ (𝑆 ⊆ (2𝑜 ↑𝑚 𝐴) → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜})))
68	7, 67	ax-mp 5	. . . . 5 ⊢ ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜}))
69		pwfi2f1o.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1𝑜}))
70	68, 69	eqtr4i 2647	. . . 4 ⊢ ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆) = 𝐹
71		f1oeq1 6127	. . . 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)))
72	70, 71	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)))
73	66, 72	bitrd 268	. 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)))
74	9, 73	mpbid 222	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116   “ cima 5117  Ord word 5722  suc csuc 5725  Fun wfun 5882  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  (class class class)co 6650   supp csupp 7295  1_𝑜c1o 7553  2_𝑜c2o 7554   ↑_𝑚 cmap 7857  Fincfn 7955   finSupp cfsupp 8275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-supp 7296  df-1o 7560  df-2o 7561  df-map 7859  df-fsupp 8276

This theorem is referenced by:  pwfi2en  37667