Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rfovcnvf1od Structured version   Visualization version   GIF version

Theorem rfovcnvf1od 37101
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) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))
rfovd.a (𝜑𝐴𝑉)
rfovd.b (𝜑𝐵𝑊)
rfovcnvf1od.f 𝐹 = (𝐴𝑂𝐵)
Assertion
Ref Expression
rfovcnvf1od (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴) ∧ 𝐹 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})))
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.
StepHypRef Expression
1 eqid 2609 . . 3 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
2 rfovd.b . . . . . . . 8 (𝜑𝐵𝑊)
3 ssrab2 3649 . . . . . . . . 9 {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵
43a1i 11 . . . . . . . 8 (𝜑 → {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵)
52, 4sselpwd 4728 . . . . . . 7 (𝜑 → {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵)
65adantr 479 . . . . . 6 ((𝜑𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵)
7 eqid 2609 . . . . . 6 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
86, 7fmptd 6276 . . . . 5 (𝜑 → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
9 pwexg 4770 . . . . . . 7 (𝐵𝑊 → 𝒫 𝐵 ∈ V)
102, 9syl 17 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
11 rfovd.a . . . . . 6 (𝜑𝐴𝑉)
1210, 11elmapd 7735 . . . . 5 (𝜑 → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ (𝒫 𝐵𝑚 𝐴) ↔ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵))
138, 12mpbird 245 . . . 4 (𝜑 → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ (𝒫 𝐵𝑚 𝐴))
1413adantr 479 . . 3 ((𝜑𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ (𝒫 𝐵𝑚 𝐴))
15 xpexg 6835 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
1611, 2, 15syl2anc 690 . . . . 5 (𝜑 → (𝐴 × 𝐵) ∈ V)
1716adantr 479 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝐴 × 𝐵) ∈ V)
1810, 11elmapd 7735 . . . . . . . . . . . 12 (𝜑 → (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵))
1918biimpa 499 . . . . . . . . . . 11 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → 𝑓:𝐴⟶𝒫 𝐵)
2019ffvelrnda 6251 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝒫 𝐵)
2120ex 448 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝑥𝐴 → (𝑓𝑥) ∈ 𝒫 𝐵))
22 elpwi 4116 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝒫 𝐵 → (𝑓𝑥) ⊆ 𝐵)
2322sseld 3566 . . . . . . . . 9 ((𝑓𝑥) ∈ 𝒫 𝐵 → (𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
2421, 23syl6 34 . . . . . . . 8 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝑥𝐴 → (𝑦 ∈ (𝑓𝑥) → 𝑦𝐵)))
2524imdistand 723 . . . . . . 7 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → (𝑥𝐴𝑦𝐵)))
26 a1tru 1490 . . . . . . . 8 ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → ⊤)
2726a1i 11 . . . . . . 7 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → ⊤))
2825, 27jcad 553 . . . . . 6 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → ((𝑥𝐴𝑦𝐵) ∧ ⊤)))
2928ssopab2dv 4918 . . . . 5 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ⊤)})
30 opabssxp 5105 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ⊤)} ⊆ (𝐴 × 𝐵)
3129, 30syl6ss 3579 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ (𝐴 × 𝐵))
3217, 31sselpwd 4728 . . 3 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ 𝒫 (𝐴 × 𝐵))
33 simplrr 796 . . . . . 6 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))
34 elmapfn 7743 . . . . . 6 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → 𝑓 Fn 𝐴)
3533, 34syl 17 . . . . 5 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 Fn 𝐴)
362ad2antrr 757 . . . . . 6 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝐵𝑊)
37 rabexg 4733 . . . . . . 7 (𝐵𝑊 → {𝑦𝐵𝑥𝑟𝑦} ∈ V)
3837ralrimivw 2949 . . . . . 6 (𝐵𝑊 → ∀𝑥𝐴 {𝑦𝐵𝑥𝑟𝑦} ∈ V)
39 nfcv 2750 . . . . . . 7 𝑥𝐴
4039fnmptf 5914 . . . . . 6 (∀𝑥𝐴 {𝑦𝐵𝑥𝑟𝑦} ∈ V → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) Fn 𝐴)
4136, 38, 403syl 18 . . . . 5 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) Fn 𝐴)
42 dfin5 3547 . . . . . . 7 (𝐵 ∩ (𝑓𝑢)) = {𝑏𝐵𝑏 ∈ (𝑓𝑢)}
43 simpllr 794 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴)))
4443simprd 477 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))
45 elmapi 7742 . . . . . . . . . . 11 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
4644, 45syl 17 . . . . . . . . . 10 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
47 simpr 475 . . . . . . . . . 10 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → 𝑢𝐴)
4846, 47ffvelrnd 6252 . . . . . . . . 9 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑓𝑢) ∈ 𝒫 𝐵)
4948elpwid 4117 . . . . . . . 8 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑓𝑢) ⊆ 𝐵)
50 sseqin2 3778 . . . . . . . 8 ((𝑓𝑢) ⊆ 𝐵 ↔ (𝐵 ∩ (𝑓𝑢)) = (𝑓𝑢))
5149, 50sylib 206 . . . . . . 7 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝐵 ∩ (𝑓𝑢)) = (𝑓𝑢))
52 ibar 523 . . . . . . . . 9 (𝑢𝐴 → (𝑏 ∈ (𝑓𝑢) ↔ (𝑢𝐴𝑏 ∈ (𝑓𝑢))))
5352rabbidv 3163 . . . . . . . 8 (𝑢𝐴 → {𝑏𝐵𝑏 ∈ (𝑓𝑢)} = {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))})
5453adantl 480 . . . . . . 7 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → {𝑏𝐵𝑏 ∈ (𝑓𝑢)} = {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))})
5542, 51, 543eqtr3a 2667 . . . . . 6 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑓𝑢) = {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))})
56 breq2 4581 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑥𝑟𝑦𝑥𝑟𝑏))
5756cbvrabv 3171 . . . . . . . . . 10 {𝑦𝐵𝑥𝑟𝑦} = {𝑏𝐵𝑥𝑟𝑏}
58 breq1 4580 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑥𝑟𝑏𝑎𝑟𝑏))
59 df-br 4578 . . . . . . . . . . . 12 (𝑎𝑟𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑟)
6058, 59syl6bb 274 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥𝑟𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑟))
6160rabbidv 3163 . . . . . . . . . 10 (𝑥 = 𝑎 → {𝑏𝐵𝑥𝑟𝑏} = {𝑏𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟})
6257, 61syl5eq 2655 . . . . . . . . 9 (𝑥 = 𝑎 → {𝑦𝐵𝑥𝑟𝑦} = {𝑏𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟})
6362cbvmptv 4672 . . . . . . . 8 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑎𝐴 ↦ {𝑏𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟})
6463a1i 11 . . . . . . 7 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑎𝐴 ↦ {𝑏𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟}))
65 simpr 475 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → 𝑎 = 𝑢)
6665opeq1d 4340 . . . . . . . . . 10 (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑏⟩)
67 simpllr 794 . . . . . . . . . 10 (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
6866, 67eleq12d 2681 . . . . . . . . 9 (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → (⟨𝑎, 𝑏⟩ ∈ 𝑟 ↔ ⟨𝑢, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
69 vex 3175 . . . . . . . . . 10 𝑢 ∈ V
70 vex 3175 . . . . . . . . . 10 𝑏 ∈ V
71 simpl 471 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑏) → 𝑥 = 𝑢)
7271eleq1d 2671 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑏) → (𝑥𝐴𝑢𝐴))
73 simpr 475 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑏) → 𝑦 = 𝑏)
7471fveq2d 6091 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑏) → (𝑓𝑥) = (𝑓𝑢))
7573, 74eleq12d 2681 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑏) → (𝑦 ∈ (𝑓𝑥) ↔ 𝑏 ∈ (𝑓𝑢)))
7672, 75anbi12d 742 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑏) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) ↔ (𝑢𝐴𝑏 ∈ (𝑓𝑢))))
7769, 70, 76opelopaba 4905 . . . . . . . . 9 (⟨𝑢, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ (𝑢𝐴𝑏 ∈ (𝑓𝑢)))
7868, 77syl6bb 274 . . . . . . . 8 (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → (⟨𝑎, 𝑏⟩ ∈ 𝑟 ↔ (𝑢𝐴𝑏 ∈ (𝑓𝑢))))
7978rabbidv 3163 . . . . . . 7 (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → {𝑏𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟} = {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))})
802ad3antrrr 761 . . . . . . . 8 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → 𝐵𝑊)
81 rabexg 4733 . . . . . . . 8 (𝐵𝑊 → {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))} ∈ V)
8280, 81syl 17 . . . . . . 7 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))} ∈ V)
8364, 79, 47, 82fvmptd 6181 . . . . . 6 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑢) = {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))})
8455, 83eqtr4d 2646 . . . . 5 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑓𝑢) = ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑢))
8535, 41, 84eqfnfvd 6206 . . . 4 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
86 simplrl 795 . . . . . . . 8 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
8786elpwid 4117 . . . . . . 7 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
88 xpss 5137 . . . . . . 7 (𝐴 × 𝐵) ⊆ (V × V)
8987, 88syl6ss 3579 . . . . . 6 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V))
90 df-rel 5034 . . . . . 6 (Rel 𝑟𝑟 ⊆ (V × V))
9189, 90sylibr 222 . . . . 5 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel 𝑟)
92 relopab 5156 . . . . . 6 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
9392a1i 11 . . . . 5 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
94 simpl 471 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
952, 94anim12i 587 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) → (𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)))
9695anim1i 589 . . . . 5 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
97 vex 3175 . . . . . . . 8 𝑣 ∈ V
98 simpl 471 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
9998eleq1d 2671 . . . . . . . . 9 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑥𝐴𝑢𝐴))
100 simpr 475 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
10198fveq2d 6091 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑓𝑥) = (𝑓𝑢))
102100, 101eleq12d 2681 . . . . . . . . 9 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑦 ∈ (𝑓𝑥) ↔ 𝑣 ∈ (𝑓𝑢)))
10399, 102anbi12d 742 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) ↔ (𝑢𝐴𝑣 ∈ (𝑓𝑢))))
10469, 97, 103opelopaba 4905 . . . . . . 7 (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ (𝑢𝐴𝑣 ∈ (𝑓𝑢)))
105 breq2 4581 . . . . . . . . . . . 12 (𝑏 = 𝑣 → (𝑢𝑟𝑏𝑢𝑟𝑣))
106 df-br 4578 . . . . . . . . . . . 12 (𝑢𝑟𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑟)
107105, 106syl6bb 274 . . . . . . . . . . 11 (𝑏 = 𝑣 → (𝑢𝑟𝑏 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑟))
108107elrab 3330 . . . . . . . . . 10 (𝑣 ∈ {𝑏𝐵𝑢𝑟𝑏} ↔ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))
109108anbi2i 725 . . . . . . . . 9 ((𝑢𝐴𝑣 ∈ {𝑏𝐵𝑢𝑟𝑏}) ↔ (𝑢𝐴 ∧ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟)))
110109a1i 11 . . . . . . . 8 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑢𝐴𝑣 ∈ {𝑏𝐵𝑢𝑟𝑏}) ↔ (𝑢𝐴 ∧ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))))
111 simplr 787 . . . . . . . . . . . 12 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
112 breq1 4580 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (𝑥𝑟𝑦𝑎𝑟𝑦))
113112rabbidv 3163 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → {𝑦𝐵𝑥𝑟𝑦} = {𝑦𝐵𝑎𝑟𝑦})
114 breq2 4581 . . . . . . . . . . . . . . 15 (𝑦 = 𝑏 → (𝑎𝑟𝑦𝑎𝑟𝑏))
115114cbvrabv 3171 . . . . . . . . . . . . . 14 {𝑦𝐵𝑎𝑟𝑦} = {𝑏𝐵𝑎𝑟𝑏}
116113, 115syl6eq 2659 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → {𝑦𝐵𝑥𝑟𝑦} = {𝑏𝐵𝑎𝑟𝑏})
117116cbvmptv 4672 . . . . . . . . . . . 12 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑎𝐴 ↦ {𝑏𝐵𝑎𝑟𝑏})
118111, 117syl6eq 2659 . . . . . . . . . . 11 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → 𝑓 = (𝑎𝐴 ↦ {𝑏𝐵𝑎𝑟𝑏}))
119 breq1 4580 . . . . . . . . . . . . 13 (𝑎 = 𝑢 → (𝑎𝑟𝑏𝑢𝑟𝑏))
120119rabbidv 3163 . . . . . . . . . . . 12 (𝑎 = 𝑢 → {𝑏𝐵𝑎𝑟𝑏} = {𝑏𝐵𝑢𝑟𝑏})
121120adantl 480 . . . . . . . . . . 11 (((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → {𝑏𝐵𝑎𝑟𝑏} = {𝑏𝐵𝑢𝑟𝑏})
122 simpr 475 . . . . . . . . . . 11 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → 𝑢𝐴)
123 rabexg 4733 . . . . . . . . . . . 12 (𝐵𝑊 → {𝑏𝐵𝑢𝑟𝑏} ∈ V)
124123ad3antrrr 761 . . . . . . . . . . 11 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → {𝑏𝐵𝑢𝑟𝑏} ∈ V)
125118, 121, 122, 124fvmptd 6181 . . . . . . . . . 10 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → (𝑓𝑢) = {𝑏𝐵𝑢𝑟𝑏})
126125eleq2d 2672 . . . . . . . . 9 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → (𝑣 ∈ (𝑓𝑢) ↔ 𝑣 ∈ {𝑏𝐵𝑢𝑟𝑏}))
127126pm5.32da 670 . . . . . . . 8 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ (𝑢𝐴𝑣 ∈ {𝑏𝐵𝑢𝑟𝑏})))
128 simplr 787 . . . . . . . . . 10 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
129128elpwid 4117 . . . . . . . . 9 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
13069, 97opeldm 5236 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ ∈ 𝑟𝑢 ∈ dom 𝑟)
131 dmss 5231 . . . . . . . . . . . . . 14 (𝑟 ⊆ (𝐴 × 𝐵) → dom 𝑟 ⊆ dom (𝐴 × 𝐵))
132 dmxpss 5469 . . . . . . . . . . . . . 14 dom (𝐴 × 𝐵) ⊆ 𝐴
133131, 132syl6ss 3579 . . . . . . . . . . . . 13 (𝑟 ⊆ (𝐴 × 𝐵) → dom 𝑟𝐴)
134133sseld 3566 . . . . . . . . . . . 12 (𝑟 ⊆ (𝐴 × 𝐵) → (𝑢 ∈ dom 𝑟𝑢𝐴))
135130, 134syl5 33 . . . . . . . . . . 11 (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟𝑢𝐴))
136135pm4.71rd 664 . . . . . . . . . 10 (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑢𝐴 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟)))
13769, 97opelrn 5264 . . . . . . . . . . . . 13 (⟨𝑢, 𝑣⟩ ∈ 𝑟𝑣 ∈ ran 𝑟)
138 rnss 5261 . . . . . . . . . . . . . . 15 (𝑟 ⊆ (𝐴 × 𝐵) → ran 𝑟 ⊆ ran (𝐴 × 𝐵))
139 rnxpss 5470 . . . . . . . . . . . . . . 15 ran (𝐴 × 𝐵) ⊆ 𝐵
140138, 139syl6ss 3579 . . . . . . . . . . . . . 14 (𝑟 ⊆ (𝐴 × 𝐵) → ran 𝑟𝐵)
141140sseld 3566 . . . . . . . . . . . . 13 (𝑟 ⊆ (𝐴 × 𝐵) → (𝑣 ∈ ran 𝑟𝑣𝐵))
142137, 141syl5 33 . . . . . . . . . . . 12 (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟𝑣𝐵))
143142pm4.71rd 664 . . . . . . . . . . 11 (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟)))
144143anbi2d 735 . . . . . . . . . 10 (𝑟 ⊆ (𝐴 × 𝐵) → ((𝑢𝐴 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟) ↔ (𝑢𝐴 ∧ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))))
145136, 144bitrd 266 . . . . . . . . 9 (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑢𝐴 ∧ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))))
146129, 145syl 17 . . . . . . . 8 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑢𝐴 ∧ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))))
147110, 127, 1463bitr4d 298 . . . . . . 7 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑟))
148104, 147syl5rbb 271 . . . . . 6 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
149148eqrelrdv2 5130 . . . . 5 (((Rel 𝑟 ∧ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ ((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
15091, 93, 96, 149syl21anc 1316 . . . 4 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
15185, 150impbida 872 . . 3 ((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) → (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
1521, 14, 32, 151f1ocnv2d 6761 . 2 (𝜑 → ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})))
153 rfovcnvf1od.f . . . 4 𝐹 = (𝐴𝑂𝐵)
154 rfovd.rf . . . . 5 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))
155154, 11, 2rfovd 37098 . . . 4 (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
156153, 155syl5eq 2655 . . 3 (𝜑𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
157 f1oeq1 6024 . . . 4 (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴)))
158 cnveq 5205 . . . . 5 (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
159158eqeq1d 2611 . . . 4 (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝐹 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})))
160157, 159anbi12d 742 . . 3 (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴) ∧ 𝐹 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})) ↔ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))))
161156, 160syl 17 . 2 (𝜑 → ((𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴) ∧ 𝐹 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})) ↔ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))))
162152, 161mpbird 245 1 (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴) ∧ 𝐹 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wtru 1475  wcel 1976  wral 2895  {crab 2899  Vcvv 3172  cin 3538  wss 3539  𝒫 cpw 4107  cop 4130   class class class wbr 4577  {copab 4636  cmpt 4637   × cxp 5025  ccnv 5026  dom cdm 5027  ran crn 5028  Rel wrel 5032   Fn wfn 5784  wf 5785  1-1-ontowf1o 5788  cfv 5789  (class class class)co 6526  cmpt2 6528  𝑚 cmap 7721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-map 7723
This theorem is referenced by:  rfovcnvd  37102  rfovf1od  37103
  Copyright terms: Public domain W3C validator