Theorem pw1map 16769

Description: Mapping between (𝒫 1_o ↑_𝑚 𝐴) and subsets of 𝐴. (Contributed by Jim Kingdon, 9-Jan-2026.)

Hypothesis

Ref	Expression
pw1map.f	⊢ 𝐹 = (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})

Assertion

Ref	Expression
pw1map	⊢ (𝐴 ∈ 𝑉 → 𝐹:(𝒫 1o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)

Distinct variable groups:   𝐴,𝑠,𝑧   𝑉,𝑠,𝑧

Allowed substitution hints:   𝐹(𝑧,𝑠)

Proof of Theorem pw1map

Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pw1map.f	. 2 ⊢ 𝐹 = (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
2		ssrab2 3323	. . . 4 ⊢ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ⊆ 𝐴
3		elpw2g 4268	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ∈ 𝒫 𝐴 ↔ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ⊆ 𝐴))
4	2, 3	mpbiri 168	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ∈ 𝒫 𝐴)
5	4	adantr 276	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (𝒫 1o ↑𝑚 𝐴)) → {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ∈ 𝒫 𝐴)
6		fmelpw1o 7557	. . . . 5 ⊢ if(𝑢 ∈ 𝑤, 1o, ∅) ∈ 𝒫 1o
7	6	a1i 9	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑤 ∈ 𝒫 𝐴) ∧ 𝑢 ∈ 𝐴) → if(𝑢 ∈ 𝑤, 1o, ∅) ∈ 𝒫 1o)
8	7	fmpttd 5832	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑤 ∈ 𝒫 𝐴) → (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)):𝐴⟶𝒫 1o)
9		1oex 6655	. . . . . 6 ⊢ 1o ∈ V
10	9	pwex 4296	. . . . 5 ⊢ 𝒫 1o ∈ V
11	10	a1i 9	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑤 ∈ 𝒫 𝐴) → 𝒫 1o ∈ V)
12		simpl 109	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑤 ∈ 𝒫 𝐴) → 𝐴 ∈ 𝑉)
13	11, 12	elmapd 6896	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑤 ∈ 𝒫 𝐴) → ((𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) ∈ (𝒫 1o ↑𝑚 𝐴) ↔ (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)):𝐴⟶𝒫 1o))
14	8, 13	mpbird 167	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑤 ∈ 𝒫 𝐴) → (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) ∈ (𝒫 1o ↑𝑚 𝐴))
15		simplr 529	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)))
16	15	fveq1d 5672	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → (𝑠‘𝑧) = ((𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))‘𝑧))
17		eqid 2232	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))
18		elequ1 2207	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
19	18	ifbid 3644	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → if(𝑢 ∈ 𝑤, 1o, ∅) = if(𝑧 ∈ 𝑤, 1o, ∅))
20		simpr 110	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
21		0ex 4237	. . . . . . . . . . 11 ⊢ ∅ ∈ V
22	9, 21	ifex 4607	. . . . . . . . . 10 ⊢ if(𝑧 ∈ 𝑤, 1o, ∅) ∈ V
23	22	a1i 9	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → if(𝑧 ∈ 𝑤, 1o, ∅) ∈ V)
24	17, 19, 20, 23	fvmptd3 5771	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))‘𝑧) = if(𝑧 ∈ 𝑤, 1o, ∅))
25	16, 24	eqtrd 2265	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → (𝑠‘𝑧) = if(𝑧 ∈ 𝑤, 1o, ∅))
26	25	eqeq1d 2241	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → ((𝑠‘𝑧) = 1o ↔ if(𝑧 ∈ 𝑤, 1o, ∅) = 1o))
27		iftrueb01 7533	. . . . . 6 ⊢ (if(𝑧 ∈ 𝑤, 1o, ∅) = 1o ↔ 𝑧 ∈ 𝑤)
28	26, 27	bitr2di 197	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝑤 ↔ (𝑠‘𝑧) = 1o))
29	28	rabbidva 2801	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) → {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤} = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
30		elpwi 3678	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝒫 𝐴 → 𝑤 ⊆ 𝐴)
31		dfss1 3425	. . . . . . . . 9 ⊢ (𝑤 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑤) = 𝑤)
32	30, 31	sylib 122	. . . . . . . 8 ⊢ (𝑤 ∈ 𝒫 𝐴 → (𝐴 ∩ 𝑤) = 𝑤)
33		dfin5 3218	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑤) = {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤}
34	32, 33	eqtr3di 2280	. . . . . . 7 ⊢ (𝑤 ∈ 𝒫 𝐴 → 𝑤 = {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤})
35	34	eqeq1d 2241	. . . . . 6 ⊢ (𝑤 ∈ 𝒫 𝐴 → (𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ↔ {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤} = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
36	35	adantl 277	. . . . 5 ⊢ ((𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴) → (𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ↔ {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤} = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
37	36	ad2antlr 489	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) → (𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ↔ {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤} = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
38	29, 37	mpbird 167	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) → 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
39		simplrl 537	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑠 ∈ (𝒫 1o ↑𝑚 𝐴))
40	10	a1i 9	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝒫 1o ∈ V)
41		simpll 527	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝐴 ∈ 𝑉)
42	40, 41	elmapd 6896	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ↔ 𝑠:𝐴⟶𝒫 1o))
43	39, 42	mpbid 147	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑠:𝐴⟶𝒫 1o)
44	43	feqmptd 5730	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑠 = (𝑢 ∈ 𝐴 ↦ (𝑠‘𝑢)))
45		simpr 110	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
46	45	eleq2d 2302	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → (𝑢 ∈ 𝑤 ↔ 𝑢 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
47		fveqeq2 5679	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ((𝑠‘𝑧) = 1o ↔ (𝑠‘𝑢) = 1o))
48	47	elrab 2973	. . . . . . . . 9 ⊢ (𝑢 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ↔ (𝑢 ∈ 𝐴 ∧ (𝑠‘𝑢) = 1o))
49	46, 48	bitrdi 196	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → (𝑢 ∈ 𝑤 ↔ (𝑢 ∈ 𝐴 ∧ (𝑠‘𝑢) = 1o)))
50	49	baibd 931	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∈ 𝑤 ↔ (𝑠‘𝑢) = 1o))
51	50	ifbid 3644	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → if(𝑢 ∈ 𝑤, 1o, ∅) = if((𝑠‘𝑢) = 1o, 1o, ∅))
52	43	ffvelcdmda 5812	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑠‘𝑢) ∈ 𝒫 1o)
53		pw1if 7535	. . . . . . 7 ⊢ ((𝑠‘𝑢) ∈ 𝒫 1o → if((𝑠‘𝑢) = 1o, 1o, ∅) = (𝑠‘𝑢))
54	52, 53	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → if((𝑠‘𝑢) = 1o, 1o, ∅) = (𝑠‘𝑢))
55	51, 54	eqtr2d 2266	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑠‘𝑢) = if(𝑢 ∈ 𝑤, 1o, ∅))
56	55	mpteq2dva 4200	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → (𝑢 ∈ 𝐴 ↦ (𝑠‘𝑢)) = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)))
57	44, 56	eqtrd 2265	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)))
58	38, 57	impbida 600	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) → (𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) ↔ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
59	1, 5, 14, 58	f1o2d 6260	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:(𝒫 1o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {crab 2524  Vcvv 2813   ∩ cin 3210   ⊆ wss 3211  ∅c0 3508  ifcif 3620  𝒫 cpw 3669   ↦ cmpt 4171  ⟶wf 5348  –1-1-onto→wf1o 5351  ‘cfv 5352  (class class class)co 6050  1_oc1o 6640   ↑_𝑚 cmap 6882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1o 6647  df-map 6884

This theorem is referenced by:  pw1mapen  16770