Theorem rfovcnvf1od 37318

Description: Properties of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 27-Apr-2021.)

Hypotheses

Ref	Expression
rfovd.rf	⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))
rfovd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rfovd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
rfovcnvf1od.f	⊢ 𝐹 = (𝐴𝑂𝐵)

Assertion

Ref	Expression
rfovcnvf1od	⊢ (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑓,𝑟,𝑥,𝑦   𝐵,𝑎,𝑏,𝑓,𝑟,𝑥,𝑦   𝑊,𝑎,𝑥   𝜑,𝑎,𝑏,𝑓,𝑟,𝑥,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦,𝑓,𝑟,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑓,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑓,𝑟,𝑎,𝑏)   𝑊(𝑦,𝑓,𝑟,𝑏)

Proof of Theorem rfovcnvf1od

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

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
2		rfovd.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		ssrab2 3650	. . . . . . . . 9 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ⊆ 𝐵
4	3	a1i 11	. . . . . . . 8 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ⊆ 𝐵)
5	2, 4	sselpwd 4734	. . . . . . 7 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵)
7		eqid 2610	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
8	6, 7	fmptd 6292	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
9		pwexg 4776	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ∈ V)
10	2, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
11		rfovd.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12	10, 11	elmapd 7758	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↔ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵))
13	8, 12	mpbird 246	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ (𝒫 𝐵 ↑𝑚 𝐴))
14	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ (𝒫 𝐵 ↑𝑚 𝐴))
15		xpexg 6858	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)
16	11, 2, 15	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
17	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → (𝐴 × 𝐵) ∈ V)
18	10, 11	elmapd 7758	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵))
19	18	biimpa 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → 𝑓:𝐴⟶𝒫 𝐵)
20	19	ffvelrnda 6267	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ 𝒫 𝐵)
21	20	ex 449	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ 𝒫 𝐵))
22		elpwi 4117	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ 𝒫 𝐵 → (𝑓‘𝑥) ⊆ 𝐵)
23	22	sseld 3567	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) ∈ 𝒫 𝐵 → (𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵))
24	21, 23	syl6 34	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → (𝑥 ∈ 𝐴 → (𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵)))
25	24	imdistand 724	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
26		a1tru 1491	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → ⊤)
27	26	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → ⊤))
28	25, 27	jcad 554	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⊤)))
29	28	ssopab2dv 4929	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⊤)})
30		opabssxp 5116	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⊤)} ⊆ (𝐴 × 𝐵)
31	29, 30	syl6ss 3580	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ⊆ (𝐴 × 𝐵))
32	17, 31	sselpwd 4734	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ∈ 𝒫 (𝐴 × 𝐵))
33		simplrr 797	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))
34		elmapfn 7766	. . . . . 6 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → 𝑓 Fn 𝐴)
35	33, 34	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 Fn 𝐴)
36	2	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝐵 ∈ 𝑊)
37		rabexg 4739	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V)
38	37	ralrimivw 2950	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V)
39		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
40	39	fnmptf 5929	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) Fn 𝐴)
41	36, 38, 40	3syl 18	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) Fn 𝐴)
42		dfin5 3548	. . . . . . 7 ⊢ (𝐵 ∩ (𝑓‘𝑢)) = {𝑏 ∈ 𝐵 ∣ 𝑏 ∈ (𝑓‘𝑢)}
43		simpllr 795	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)))
44	43	simprd 478	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))
45		elmapi 7765	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
46	44, 45	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
47		simpr 476	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
48	46, 47	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ∈ 𝒫 𝐵)
49	48	elpwid 4118	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ⊆ 𝐵)
50		sseqin2 3779	. . . . . . . 8 ⊢ ((𝑓‘𝑢) ⊆ 𝐵 ↔ (𝐵 ∩ (𝑓‘𝑢)) = (𝑓‘𝑢))
51	49, 50	sylib 207	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝐵 ∩ (𝑓‘𝑢)) = (𝑓‘𝑢))
52		ibar 524	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐴 → (𝑏 ∈ (𝑓‘𝑢) ↔ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))))
53	52	rabbidv 3164	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → {𝑏 ∈ 𝐵 ∣ 𝑏 ∈ (𝑓‘𝑢)} = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))})
54	53	adantl 481	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → {𝑏 ∈ 𝐵 ∣ 𝑏 ∈ (𝑓‘𝑢)} = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))})
55	42, 51, 54	3eqtr3a 2668	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))})
56		breq2 4587	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝑥𝑟𝑦 ↔ 𝑥𝑟𝑏))
57	56	cbvrabv 3172	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑏 ∈ 𝐵 ∣ 𝑥𝑟𝑏}
58		breq1 4586	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑥𝑟𝑏 ↔ 𝑎𝑟𝑏))
59		df-br 4584	. . . . . . . . . . . 12 ⊢ (𝑎𝑟𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑟)
60	58, 59	syl6bb 275	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥𝑟𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑟))
61	60	rabbidv 3164	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → {𝑏 ∈ 𝐵 ∣ 𝑥𝑟𝑏} = {𝑏 ∈ 𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟})
62	57, 61	syl5eq 2656	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑏 ∈ 𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟})
63	62	cbvmptv 4678	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑎 ∈ 𝐴 ↦ {𝑏 ∈ 𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟})
64	63	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑎 ∈ 𝐴 ↦ {𝑏 ∈ 𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟}))
65		simpr 476	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → 𝑎 = 𝑢)
66	65	opeq1d 4346	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑏⟩)
67		simpllr 795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
68	66, 67	eleq12d 2682	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → (⟨𝑎, 𝑏⟩ ∈ 𝑟 ↔ ⟨𝑢, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))
69		vex 3176	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
70		vex 3176	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
71		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑢)
72	71	eleq1d 2672	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → (𝑥 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴))
73		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏)
74	71	fveq2d 6107	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → (𝑓‘𝑥) = (𝑓‘𝑢))
75	73, 74	eleq12d 2682	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑏 ∈ (𝑓‘𝑢)))
76	72, 75	anbi12d 743	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) ↔ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))))
77	69, 70, 76	opelopaba 4916	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ↔ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢)))
78	68, 77	syl6bb 275	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → (⟨𝑎, 𝑏⟩ ∈ 𝑟 ↔ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))))
79	78	rabbidv 3164	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → {𝑏 ∈ 𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟} = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))})
80	2	ad3antrrr 762	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → 𝐵 ∈ 𝑊)
81		rabexg 4739	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))} ∈ V)
82	80, 81	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))} ∈ V)
83	64, 79, 47, 82	fvmptd 6197	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑢) = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))})
84	55, 83	eqtr4d 2647	. . . . 5 ⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) = ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑢))
85	35, 41, 84	eqfnfvd 6222	. . . 4 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
86		simplrl 796	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
87	86	elpwid 4118	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
88		xpss 5149	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
89	87, 88	syl6ss 3580	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V))
90		df-rel 5045	. . . . . 6 ⊢ (Rel 𝑟 ↔ 𝑟 ⊆ (V × V))
91	89, 90	sylibr 223	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel 𝑟)
92		relopab 5169	. . . . . 6 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}
93	92	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
94		simpl 472	. . . . . . 7 ⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
95	2, 94	anim12i 588	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) → (𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)))
96	95	anim1i 590	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
97		vex 3176	. . . . . . . 8 ⊢ 𝑣 ∈ V
98		simpl 472	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
99	98	eleq1d 2672	. . . . . . . . 9 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴))
100		simpr 476	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
101	98	fveq2d 6107	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑓‘𝑥) = (𝑓‘𝑢))
102	100, 101	eleq12d 2682	. . . . . . . . 9 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑣 ∈ (𝑓‘𝑢)))
103	99, 102	anbi12d 743	. . . . . . . 8 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))))
104	69, 97, 103	opelopaba 4916	. . . . . . 7 ⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢)))
105		breq2 4587	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑣 → (𝑢𝑟𝑏 ↔ 𝑢𝑟𝑣))
106		df-br 4584	. . . . . . . . . . . 12 ⊢ (𝑢𝑟𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑟)
107	105, 106	syl6bb 275	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑣 → (𝑢𝑟𝑏 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑟))
108	107	elrab 3331	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏} ↔ (𝑣 ∈ 𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))
109	108	anbi2i 726	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏}) ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟)))
110	109	a1i 11	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏}) ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))))
111		simplr 788	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
112		breq1 4586	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑥𝑟𝑦 ↔ 𝑎𝑟𝑦))
113	112	rabbidv 3164	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑎𝑟𝑦})
114		breq2 4587	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (𝑎𝑟𝑦 ↔ 𝑎𝑟𝑏))
115	114	cbvrabv 3172	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑎𝑟𝑦} = {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏}
116	113, 115	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏})
117	116	cbvmptv 4678	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑎 ∈ 𝐴 ↦ {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏})
118	111, 117	syl6eq 2660	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → 𝑓 = (𝑎 ∈ 𝐴 ↦ {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏}))
119		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑢 → (𝑎𝑟𝑏 ↔ 𝑢𝑟𝑏))
120	119	rabbidv 3164	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑢 → {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏} = {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏})
121	120	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏} = {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏})
122		simpr 476	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
123		rabexg 4739	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑊 → {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏} ∈ V)
124	123	ad3antrrr 762	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏} ∈ V)
125	118, 121, 122, 124	fvmptd 6197	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) = {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏})
126	125	eleq2d 2673	. . . . . . . . 9 ⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → (𝑣 ∈ (𝑓‘𝑢) ↔ 𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏}))
127	126	pm5.32da 671	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢)) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏})))
128		simplr 788	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
129	128	elpwid 4118	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
130	69, 97	opeldm 5250	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ 𝑟 → 𝑢 ∈ dom 𝑟)
131		dmss 5245	. . . . . . . . . . . . . 14 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → dom 𝑟 ⊆ dom (𝐴 × 𝐵))
132		dmxpss 5484	. . . . . . . . . . . . . 14 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
133	131, 132	syl6ss 3580	. . . . . . . . . . . . 13 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → dom 𝑟 ⊆ 𝐴)
134	133	sseld 3567	. . . . . . . . . . . 12 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (𝑢 ∈ dom 𝑟 → 𝑢 ∈ 𝐴))
135	130, 134	syl5 33	. . . . . . . . . . 11 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 → 𝑢 ∈ 𝐴))
136	135	pm4.71rd 665	. . . . . . . . . 10 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑢 ∈ 𝐴 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟)))
137	69, 97	opelrn 5278	. . . . . . . . . . . . 13 ⊢ (⟨𝑢, 𝑣⟩ ∈ 𝑟 → 𝑣 ∈ ran 𝑟)
138		rnss 5275	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → ran 𝑟 ⊆ ran (𝐴 × 𝐵))
139		rnxpss 5485	. . . . . . . . . . . . . . 15 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
140	138, 139	syl6ss 3580	. . . . . . . . . . . . . 14 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → ran 𝑟 ⊆ 𝐵)
141	140	sseld 3567	. . . . . . . . . . . . 13 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (𝑣 ∈ ran 𝑟 → 𝑣 ∈ 𝐵))
142	137, 141	syl5 33	. . . . . . . . . . . 12 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 → 𝑣 ∈ 𝐵))
143	142	pm4.71rd 665	. . . . . . . . . . 11 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑣 ∈ 𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟)))
144	143	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → ((𝑢 ∈ 𝐴 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟) ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))))
145	136, 144	bitrd 267	. . . . . . . . 9 ⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))))
146	129, 145	syl 17	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))))
147	110, 127, 146	3bitr4d 299	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢)) ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑟))
148	104, 147	syl5rbb 272	. . . . . 6 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))
149	148	eqrelrdv2 5142	. . . . 5 ⊢ (((Rel 𝑟 ∧ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ ((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
150	91, 93, 96, 149	syl21anc 1317	. . . 4 ⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
151	85, 150	impbida 873	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴))) → (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
152	1, 14, 32, 151	f1ocnv2d 6784	. 2 ⊢ (𝜑 → ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴) ∧ ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})))
153		rfovcnvf1od.f	. . . 4 ⊢ 𝐹 = (𝐴𝑂𝐵)
154		rfovd.rf	. . . . 5 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))
155	154, 11, 2	rfovd 37315	. . . 4 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
156	153, 155	syl5eq 2656	. . 3 ⊢ (𝜑 → 𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
157		f1oeq1 6040	. . . 4 ⊢ (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴)))
158		cnveq 5218	. . . . 5 ⊢ (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ◡𝐹 = ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
159	158	eqeq1d 2612	. . . 4 ⊢ (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ↔ ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})))
160	157, 159	anbi12d 743	. . 3 ⊢ (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) ↔ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴) ∧ ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))))
161	156, 160	syl 17	. 2 ⊢ (𝜑 → ((𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) ↔ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴) ∧ ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))))
162	152, 161	mpbird 246	1 ⊢ (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑_𝑚 cmap 7744

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-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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-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-map 7746

This theorem is referenced by:  rfovcnvd  37319  rfovf1od  37320