Theorem pw1map 16939

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 3333	. . . 4 ⊢ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ⊆ 𝐴
3		elpw2g 4287	. . . 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 7596	. . . . 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 5854	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑤 ∈ 𝒫 𝐴) → (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)):𝐴⟶𝒫 1o)
9		1oex 6685	. . . . . 6 ⊢ 1o ∈ V
10	9	pwex 4315	. . . . 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 6926	. . 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 533	. . . . . . . . 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 5692	. . . . . . . 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 2238	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))
18		elequ1 2213	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
19	18	ifbid 3659	. . . . . . . . 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 4255	. . . . . . . . . . 11 ⊢ ∅ ∈ V
22	9, 21	ifex 4627	. . . . . . . . . 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 5793	. . . . . . . 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 2271	. . . . . . 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 2247	. . . . . 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 7572	. . . . . 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 2809	. . . 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 3694	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝒫 𝐴 → 𝑤 ⊆ 𝐴)
31		dfss1 3435	. . . . . . . . 9 ⊢ (𝑤 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑤) = 𝑤)
32	30, 31	sylib 122	. . . . . . . 8 ⊢ (𝑤 ∈ 𝒫 𝐴 → (𝐴 ∩ 𝑤) = 𝑤)
33		dfin5 3227	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑤) = {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤}
34	32, 33	eqtr3di 2286	. . . . . . 7 ⊢ (𝑤 ∈ 𝒫 𝐴 → 𝑤 = {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤})
35	34	eqeq1d 2247	. . . . . 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 493	. . . 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 541	. . . . . 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 531	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ∧ 𝑤 ∈ 𝒫 𝐴)) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝐴 ∈ 𝑉)
42	40, 41	elmapd 6926	. . . . . 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 5750	. . . 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 2308	. . . . . . . . 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 5699	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ((𝑠‘𝑧) = 1o ↔ (𝑠‘𝑢) = 1o))
48	47	elrab 2982	. . . . . . . . 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 935	. . . . . . 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 3659	. . . . . 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 5834	. . . . . . 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 7574	. . . . . . 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 2272	. . . . 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 4216	. . . 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 2271	. . 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 604	. 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 6285	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:(𝒫 1o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  ifcif 3635  𝒫 cpw 3685   ↦ cmpt 4187  ⟶wf 5368  –1-1-onto→wf1o 5371  ‘cfv 5372  (class class class)co 6075  1_oc1o 6670   ↑_𝑚 cmap 6912

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1o 6677  df-map 6914

This theorem is referenced by:  pw1mapen  16940