Theorem pw1map 16532

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 3310	. . . 4 ⊢ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ⊆ 𝐴
3		elpw2g 4244	. . . 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 7455	. . . . 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 5798	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑤 ∈ 𝒫 𝐴) → (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)):𝐴⟶𝒫 1o)
9		1oex 6585	. . . . . 6 ⊢ 1o ∈ V
10	9	pwex 4271	. . . . 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 6826	. . 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 528	. . . . . . . . 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 5637	. . . . . . . 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 2229	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))
18		elequ1 2204	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
19	18	ifbid 3625	. . . . . . . . 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 4214	. . . . . . . . . . 11 ⊢ ∅ ∈ V
22	9, 21	ifex 4581	. . . . . . . . . 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 5736	. . . . . . . 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 2262	. . . . . . 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 2238	. . . . . 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 7431	. . . . . 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 2788	. . . 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 3659	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝒫 𝐴 → 𝑤 ⊆ 𝐴)
31		dfss1 3409	. . . . . . . . 9 ⊢ (𝑤 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑤) = 𝑤)
32	30, 31	sylib 122	. . . . . . . 8 ⊢ (𝑤 ∈ 𝒫 𝐴 → (𝐴 ∩ 𝑤) = 𝑤)
33		dfin5 3205	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑤) = {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤}
34	32, 33	eqtr3di 2277	. . . . . . 7 ⊢ (𝑤 ∈ 𝒫 𝐴 → 𝑤 = {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤})
35	34	eqeq1d 2238	. . . . . 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 535	. . . . . 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 6826	. . . . . 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 5695	. . . 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 2299	. . . . . . . . 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 5644	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ((𝑠‘𝑧) = 1o ↔ (𝑠‘𝑢) = 1o))
48	47	elrab 2960	. . . . . . . . 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 928	. . . . . . 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 3625	. . . . . 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 5778	. . . . . . 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 7433	. . . . . . 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 2263	. . . . 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 4177	. . . 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 2262	. . 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 598	. 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 6223	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:(𝒫 1o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {crab 2512  Vcvv 2800   ∩ cin 3197   ⊆ wss 3198  ∅c0 3492  ifcif 3603  𝒫 cpw 3650   ↦ cmpt 4148  ⟶wf 5320  –1-1-onto→wf1o 5323  ‘cfv 5324  (class class class)co 6013  1_oc1o 6570   ↑_𝑚 cmap 6812

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1o 6577  df-map 6814

This theorem is referenced by:  pw1mapen  16533