Theorem fpwrelmap 30970

Description: Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 10135 and marypha2lem1 9124. (Contributed by Thierry Arnoux, 28-Aug-2017.)

Hypotheses

Ref	Expression
fpwrelmap.1	⊢ 𝐴 ∈ V
fpwrelmap.2	⊢ 𝐵 ∈ V
fpwrelmap.3	⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})

Assertion

Ref	Expression
fpwrelmap	⊢ 𝑀:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)

Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦

Allowed substitution hints:   𝑀(𝑥,𝑦,𝑓)

Proof of Theorem fpwrelmap

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fpwrelmap.3	. . 3 ⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
2		fpwrelmap.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	a1i 11	. . . . 5 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝐴 ∈ V)
4		simpr 484	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑦 ∈ (𝑓‘𝑥))
5		elmapi 8595	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
6	5	ffvelrnda 6943	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ 𝒫 𝐵)
7	6	adantr 480	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑓‘𝑥) ∈ 𝒫 𝐵)
8		elelpwi 4542	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ∈ 𝒫 𝐵) → 𝑦 ∈ 𝐵)
9	4, 7, 8	syl2anc 583	. . . . . . . 8 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑦 ∈ 𝐵)
10	9	ex 412	. . . . . . 7 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵))
11	10	alrimiv 1931	. . . . . 6 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵))
12		abss 3990	. . . . . . 7 ⊢ ({𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵))
13		fpwrelmap.2	. . . . . . . 8 ⊢ 𝐵 ∈ V
14	13	ssex 5240	. . . . . . 7 ⊢ ({𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐵 → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ V)
15	12, 14	sylbir 234	. . . . . 6 ⊢ (∀𝑦(𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ V)
16	11, 15	syl 17	. . . . 5 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ V)
17	3, 16	opabex3d 7781	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ∈ V)
18	17	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ∈ V)
19	2	mptex 7081	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ V
20	19	a1i 11	. . 3 ⊢ ((⊤ ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ V)
21	10	imdistanda 571	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
22	21	ssopab2dv 5457	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)})
23	22	adantr 480	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)})
24		simpr 484	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
25		df-xp 5586	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)})
27	23, 24, 26	3sstr4d 3964	. . . . . . 7 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 ⊆ (𝐴 × 𝐵))
28		velpw 4535	. . . . . . 7 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑟 ⊆ (𝐴 × 𝐵))
29	27, 28	sylibr 233	. . . . . 6 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
30	5	feqmptd 6819	. . . . . . . 8 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)))
31	30	adantr 480	. . . . . . 7 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)))
32		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)
33		nfopab1 5140	. . . . . . . . . 10 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}
34	33	nfeq2 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}
35	32, 34	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
36		df-rab 3072	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)}
37	36	a1i 11	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)})
38		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)
39		nfopab2 5141	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}
40	39	nfeq2 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}
41	38, 40	nfan 1903	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
42		nfv 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
43	41, 42	nfan 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑦((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴)
44	9	adantllr 715	. . . . . . . . . . . . 13 ⊢ ((((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑦 ∈ 𝐵)
45		df-br 5071	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
46		eleq2 2827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))
47		opabidw 5431	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))
48	46, 47	bitrdi 286	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))))
49	45, 48	syl5bb 282	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} → (𝑥𝑟𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))))
50	49	ad2antlr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑥𝑟𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))))
51		elfvdm 6788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑓‘𝑥) → 𝑥 ∈ dom 𝑓)
52	51	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥 ∈ dom 𝑓)
53	5	fdmd 6595	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → dom 𝑓 = 𝐴)
54	53	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → dom 𝑓 = 𝐴)
55	52, 54	eleqtrd 2841	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥 ∈ 𝐴)
56	55	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → (𝑦 ∈ (𝑓‘𝑥) → 𝑥 ∈ 𝐴))
57	56	pm4.71rd 562	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))))
58	57	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))))
59	50, 58	bitr4d 281	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑥𝑟𝑦 ↔ 𝑦 ∈ (𝑓‘𝑥)))
60	59	biimpar 477	. . . . . . . . . . . . 13 ⊢ ((((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥𝑟𝑦)
61	44, 60	jca 511	. . . . . . . . . . . 12 ⊢ ((((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))
62	61	ex 412	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) → (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)))
63	59	biimpd 228	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑥𝑟𝑦 → 𝑦 ∈ (𝑓‘𝑥)))
64	63	adantld 490	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓‘𝑥)))
65	62, 64	impbid 211	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)))
66	43, 65	abbid 2810	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)})
67		abid2 2881	. . . . . . . . . 10 ⊢ {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} = (𝑓‘𝑥)
68	67	a1i 11	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} = (𝑓‘𝑥))
69	37, 66, 68	3eqtr2rd 2785	. . . . . . . 8 ⊢ (((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
70	35, 69	mpteq2da 5168	. . . . . . 7 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
71	31, 70	eqtrd 2778	. . . . . 6 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
72	29, 71	jca 511	. . . . 5 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
73		ssrab2 4009	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ⊆ 𝐵
74	13, 73	elpwi2 5265	. . . . . . . . . . 11 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵
75	74	a1i 11	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵)
76	75	fmpttd 6971	. . . . . . . . 9 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
77	76	adantr 480	. . . . . . . 8 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
78		simpr 484	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
79	78	feq1d 6569	. . . . . . . 8 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑓:𝐴⟶𝒫 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵))
80	77, 79	mpbird 256	. . . . . . 7 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑓:𝐴⟶𝒫 𝐵)
81	13	pwex 5298	. . . . . . . 8 ⊢ 𝒫 𝐵 ∈ V
82	81, 2	elmap 8617	. . . . . . 7 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
83	80, 82	sylibr 233	. . . . . 6 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))
84		elpwi 4539	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → 𝑟 ⊆ (𝐴 × 𝐵))
85	84	adantr 480	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
86		xpss 5596	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
87	85, 86	sstrdi 3929	. . . . . . . 8 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V))
88		df-rel 5587	. . . . . . . 8 ⊢ (Rel 𝑟 ↔ 𝑟 ⊆ (V × V))
89	87, 88	sylibr 233	. . . . . . 7 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel 𝑟)
90		relopabv 5720	. . . . . . . 8 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}
91	90	a1i 11	. . . . . . 7 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
92		id 22	. . . . . . 7 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
93		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
94		nfmpt1 5178	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
95	94	nfeq2 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
96	93, 95	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
97		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
98	42	nfci 2889	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐴
99		nfrab1 3310	. . . . . . . . . . 11 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}
100	98, 99	nfmpt 5177	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
101	100	nfeq2 2923	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
102	97, 101	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
103		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑥𝑟
104		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑦𝑟
105		brelg 30850	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
106	84, 105	sylan 579	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
107	106	adantlr 711	. . . . . . . . . . . . . 14 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
108	107	simpld 494	. . . . . . . . . . . . 13 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥 ∈ 𝐴)
109	107	simprd 495	. . . . . . . . . . . . . 14 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ 𝐵)
110		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝑟𝑦)
111	78	fveq1d 6758	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑓‘𝑥) = ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑥))
112	13	rabex 5251	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V
113		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
114	113	fvmpt2 6868	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝐴 ∧ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V) → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
115	112, 114	mpan2 687	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
116	111, 115	sylan9eq 2799	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
117	116	eleq2d 2824	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
118		rabid 3304	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))
119	117, 118	bitrdi 286	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)))
120	108, 119	syldan 590	. . . . . . . . . . . . . 14 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)))
121	109, 110, 120	mpbir2and 709	. . . . . . . . . . . . 13 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓‘𝑥))
122	108, 121	jca 511	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))
123	122	ex 412	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑥𝑟𝑦 → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))))
124	119	simplbda 499	. . . . . . . . . . . 12 ⊢ ((((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥𝑟𝑦)
125	124	expl 457	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥𝑟𝑦))
126	123, 125	impbid 211	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑥𝑟𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))))
127	45, 126	bitr3id 284	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))))
128	127, 47	bitr4di 288	. . . . . . . 8 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))
129	96, 102, 103, 104, 33, 39, 128	eqrelrd2 30857	. . . . . . 7 ⊢ (((Rel 𝑟 ∧ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
130	89, 91, 92, 129	syl21anc 834	. . . . . 6 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
131	83, 130	jca 511	. . . . 5 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))
132	72, 131	impbii 208	. . . 4 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
133	132	a1i 11	. . 3 ⊢ (⊤ → ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))))
134	1, 18, 20, 133	f1od 7499	. 2 ⊢ (⊤ → 𝑀:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵))
135	134	mptru 1546	1 ⊢ 𝑀:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539  ⊤wtru 1540   ∈ wcel 2108  {cab 2715  {crab 3067  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  ⟨cop 4564   class class class wbr 5070  {copab 5132   ↦ cmpt 5153   × cxp 5578  dom cdm 5580  Rel wrel 5585  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575

This theorem is referenced by:  fpwrelmapffs  30971