Theorem pwfi2f1o 36684

Description: The pw2f1o 7950 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 2610	. . . . 5 ⊢ (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))
2	1	pw2f1o2 36623	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})):(2𝑜 ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3		f1of1 6049	. . . 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 3650	. . . 4 ⊢ {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2𝑜 ↑𝑚 𝐴)
7	5, 6	eqsstri 3598	. . 3 ⊢ 𝑆 ⊆ (2𝑜 ↑𝑚 𝐴)
8		f1ores 6064	. . 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 693	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆))
10		elmapfun 7767	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → Fun 𝑦)
11		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → 𝑦 ∈ (2𝑜 ↑𝑚 𝐴))
12		0ex 4718	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → ∅ ∈ V)
14	10, 11, 13	3jca 1235	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → (Fun 𝑦 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∧ ∅ ∈ V))
15	14	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (Fun 𝑦 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∧ ∅ ∈ V))
16		funisfsupp 8163	. . . . . . . . . . 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 591	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝐴 ∈ 𝑉 ∧ ∅ ∈ V))
19		elmapi 7765	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) → 𝑦:𝐴⟶2𝑜)
20	19	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → 𝑦:𝐴⟶2𝑜)
21		frnsuppeq 7194	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑦:𝐴⟶2𝑜 → (𝑦 supp ∅) = (◡𝑦 “ (2𝑜 ∖ {∅}))))
22	18, 20, 21	sylc 63	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 supp ∅) = (◡𝑦 “ (2𝑜 ∖ {∅})))
23		df-2o 7448	. . . . . . . . . . . . . . . 16 ⊢ 2𝑜 = suc 1𝑜
24		df-suc 5646	. . . . . . . . . . . . . . . . 17 ⊢ suc 1𝑜 = (1𝑜 ∪ {1𝑜})
25	24	equncomi 3721	. . . . . . . . . . . . . . . 16 ⊢ suc 1𝑜 = ({1𝑜} ∪ 1𝑜)
26	23, 25	eqtri 2632	. . . . . . . . . . . . . . 15 ⊢ 2𝑜 = ({1𝑜} ∪ 1𝑜)
27		df1o2 7459	. . . . . . . . . . . . . . . 16 ⊢ 1𝑜 = {∅}
28	27	eqcomi 2619	. . . . . . . . . . . . . . 15 ⊢ {∅} = 1𝑜
29	26, 28	difeq12i 3688	. . . . . . . . . . . . . 14 ⊢ (2𝑜 ∖ {∅}) = (({1𝑜} ∪ 1𝑜) ∖ 1𝑜)
30		difun2 4000	. . . . . . . . . . . . . . 15 ⊢ (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = ({1𝑜} ∖ 1𝑜)
31		incom 3767	. . . . . . . . . . . . . . . . 17 ⊢ ({1𝑜} ∩ 1𝑜) = (1𝑜 ∩ {1𝑜})
32		1on 7454	. . . . . . . . . . . . . . . . . . 19 ⊢ 1𝑜 ∈ On
33	32	onordi 5749	. . . . . . . . . . . . . . . . . 18 ⊢ Ord 1𝑜
34		orddisj 5679	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord 1𝑜 → (1𝑜 ∩ {1𝑜}) = ∅)
35	33, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1𝑜 ∩ {1𝑜}) = ∅
36	31, 35	eqtri 2632	. . . . . . . . . . . . . . . 16 ⊢ ({1𝑜} ∩ 1𝑜) = ∅
37		disj3 3973	. . . . . . . . . . . . . . . 16 ⊢ (({1𝑜} ∩ 1𝑜) = ∅ ↔ {1𝑜} = ({1𝑜} ∖ 1𝑜))
38	36, 37	mpbi 219	. . . . . . . . . . . . . . 15 ⊢ {1𝑜} = ({1𝑜} ∖ 1𝑜)
39	30, 38	eqtr4i 2635	. . . . . . . . . . . . . 14 ⊢ (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = {1𝑜}
40	29, 39	eqtri 2632	. . . . . . . . . . . . 13 ⊢ (2𝑜 ∖ {∅}) = {1𝑜}
41	40	imaeq2i 5383	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ (2𝑜 ∖ {∅})) = (◡𝑦 “ {1𝑜})
42	22, 41	syl6eq 2660	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 supp ∅) = (◡𝑦 “ {1𝑜}))
43	42	eleq1d 2672	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (◡𝑦 “ {1𝑜}) ∈ Fin))
44		cnvimass 5404	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ {1𝑜}) ⊆ dom 𝑦
45		fdm 5964	. . . . . . . . . . . . 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 3616	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (◡𝑦 “ {1𝑜}) ⊆ 𝐴)
48	47	biantrurd 528	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → ((◡𝑦 “ {1𝑜}) ∈ Fin ↔ ((◡𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (◡𝑦 “ {1𝑜}) ∈ Fin)))
49	17, 43, 48	3bitrd 293	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ ((◡𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (◡𝑦 “ {1𝑜}) ∈ Fin)))
50		elfpw 8151	. . . . . . . . 9 ⊢ ((◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((◡𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (◡𝑦 “ {1𝑜}) ∈ Fin))
51	49, 50	syl6bbr 277	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2𝑜 ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)))
52	51	rabbidva 3163	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ (◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)})
53		cnveq 5218	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
54	53	imaeq1d 5384	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (◡𝑥 “ {1𝑜}) = (◡𝑦 “ {1𝑜}))
55	54	cbvmptv 4678	. . . . . . . 8 ⊢ (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) = (𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑦 “ {1𝑜}))
56	55	mptpreima 5545	. . . . . . 7 ⊢ (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2𝑜 ↑𝑚 𝐴) ∣ (◡𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)}
57	52, 5, 56	3eqtr4g 2669	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝑆 = (◡(𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)))
58	57	imaeq2d 5385	. . . . 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 6057	. . . . . . 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 3795	. . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
62		foimacnv 6067	. . . . . 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 693	. . . . 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 2644	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
65		f1oeq3 6042	. . . 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 5369	. . . . . 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 2635	. . . 4 ⊢ ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜})) ↾ 𝑆) = 𝐹
71		f1oeq1 6040	. . . 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 267	. 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 221	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041  Ord word 5639  suc csuc 5642  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  (class class class)co 6549   supp csupp 7182  1_𝑜c1o 7440  2_𝑜c2o 7441   ↑_𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-supp 7183  df-1o 7447  df-2o 7448  df-map 7746  df-fsupp 8159

This theorem is referenced by:  pwfi2en  36685