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Theorem rfovcnvf1od 38615
 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 2651 . . 3 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
2 rfovd.b . . . . . . . 8 (𝜑𝐵𝑊)
3 ssrab2 3720 . . . . . . . . 9 {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵
43a1i 11 . . . . . . . 8 (𝜑 → {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵)
52, 4sselpwd 4840 . . . . . . 7 (𝜑 → {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵)
65adantr 480 . . . . . 6 ((𝜑𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵)
7 eqid 2651 . . . . . 6 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
86, 7fmptd 6425 . . . . 5 (𝜑 → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
9 pwexg 4880 . . . . . . 7 (𝐵𝑊 → 𝒫 𝐵 ∈ V)
102, 9syl 17 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
11 rfovd.a . . . . . 6 (𝜑𝐴𝑉)
1210, 11elmapd 7913 . . . . 5 (𝜑 → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ (𝒫 𝐵𝑚 𝐴) ↔ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵))
138, 12mpbird 247 . . . 4 (𝜑 → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ (𝒫 𝐵𝑚 𝐴))
1413adantr 480 . . 3 ((𝜑𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ (𝒫 𝐵𝑚 𝐴))
15 xpexg 7002 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
1611, 2, 15syl2anc 694 . . . . 5 (𝜑 → (𝐴 × 𝐵) ∈ V)
1716adantr 480 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝐴 × 𝐵) ∈ V)
1810, 11elmapd 7913 . . . . . . . . . . . 12 (𝜑 → (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵))
1918biimpa 500 . . . . . . . . . . 11 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → 𝑓:𝐴⟶𝒫 𝐵)
2019ffvelrnda 6399 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝒫 𝐵)
2120ex 449 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝑥𝐴 → (𝑓𝑥) ∈ 𝒫 𝐵))
22 elpwi 4201 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝒫 𝐵 → (𝑓𝑥) ⊆ 𝐵)
2322sseld 3635 . . . . . . . . 9 ((𝑓𝑥) ∈ 𝒫 𝐵 → (𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
2421, 23syl6 35 . . . . . . . 8 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝑥𝐴 → (𝑦 ∈ (𝑓𝑥) → 𝑦𝐵)))
2524imdistand 728 . . . . . . 7 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → (𝑥𝐴𝑦𝐵)))
26 a1tru 1540 . . . . . . . 8 ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → ⊤)
2726a1i 11 . . . . . . 7 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → ⊤))
2825, 27jcad 554 . . . . . 6 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → ((𝑥𝐴𝑦𝐵) ∧ ⊤)))
2928ssopab2dv 5033 . . . . 5 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ⊤)})
30 opabssxp 5227 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ ⊤)} ⊆ (𝐴 × 𝐵)
3129, 30syl6ss 3648 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ (𝐴 × 𝐵))
3217, 31sselpwd 4840 . . 3 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ 𝒫 (𝐴 × 𝐵))
33 simplrr 818 . . . . . 6 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))
34 elmapfn 7922 . . . . . 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 762 . . . . . 6 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝐵𝑊)
37 rabexg 4844 . . . . . . 7 (𝐵𝑊 → {𝑦𝐵𝑥𝑟𝑦} ∈ V)
3837ralrimivw 2996 . . . . . 6 (𝐵𝑊 → ∀𝑥𝐴 {𝑦𝐵𝑥𝑟𝑦} ∈ V)
39 nfcv 2793 . . . . . . 7 𝑥𝐴
4039fnmptf 6054 . . . . . 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 3615 . . . . . . 7 (𝐵 ∩ (𝑓𝑢)) = {𝑏𝐵𝑏 ∈ (𝑓𝑢)}
43 simpllr 815 . . . . . . . . . . . 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 478 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))
45 elmapi 7921 . . . . . . . . . . 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 476 . . . . . . . . . 10 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → 𝑢𝐴)
4846, 47ffvelrnd 6400 . . . . . . . . 9 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑓𝑢) ∈ 𝒫 𝐵)
4948elpwid 4203 . . . . . . . 8 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑓𝑢) ⊆ 𝐵)
50 sseqin2 3850 . . . . . . . 8 ((𝑓𝑢) ⊆ 𝐵 ↔ (𝐵 ∩ (𝑓𝑢)) = (𝑓𝑢))
5149, 50sylib 208 . . . . . . 7 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝐵 ∩ (𝑓𝑢)) = (𝑓𝑢))
52 ibar 524 . . . . . . . . 9 (𝑢𝐴 → (𝑏 ∈ (𝑓𝑢) ↔ (𝑢𝐴𝑏 ∈ (𝑓𝑢))))
5352rabbidv 3220 . . . . . . . 8 (𝑢𝐴 → {𝑏𝐵𝑏 ∈ (𝑓𝑢)} = {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))})
5453adantl 481 . . . . . . 7 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → {𝑏𝐵𝑏 ∈ (𝑓𝑢)} = {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))})
5542, 51, 543eqtr3a 2709 . . . . . 6 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑓𝑢) = {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))})
56 breq2 4689 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑥𝑟𝑦𝑥𝑟𝑏))
5756cbvrabv 3230 . . . . . . . . . 10 {𝑦𝐵𝑥𝑟𝑦} = {𝑏𝐵𝑥𝑟𝑏}
58 breq1 4688 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑥𝑟𝑏𝑎𝑟𝑏))
59 df-br 4686 . . . . . . . . . . . 12 (𝑎𝑟𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑟)
6058, 59syl6bb 276 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥𝑟𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑟))
6160rabbidv 3220 . . . . . . . . . 10 (𝑥 = 𝑎 → {𝑏𝐵𝑥𝑟𝑏} = {𝑏𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟})
6257, 61syl5eq 2697 . . . . . . . . 9 (𝑥 = 𝑎 → {𝑦𝐵𝑥𝑟𝑦} = {𝑏𝐵 ∣ ⟨𝑎, 𝑏⟩ ∈ 𝑟})
6362cbvmptv 4783 . . . . . . . 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 476 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → 𝑎 = 𝑢)
6665opeq1d 4439 . . . . . . . . . 10 (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑏⟩)
67 simpllr 815 . . . . . . . . . 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 2724 . . . . . . . . 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 3234 . . . . . . . . . 10 𝑢 ∈ V
70 vex 3234 . . . . . . . . . 10 𝑏 ∈ V
71 simpl 472 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑏) → 𝑥 = 𝑢)
7271eleq1d 2715 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑏) → (𝑥𝐴𝑢𝐴))
73 simpr 476 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑏) → 𝑦 = 𝑏)
7471fveq2d 6233 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑏) → (𝑓𝑥) = (𝑓𝑢))
7573, 74eleq12d 2724 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑏) → (𝑦 ∈ (𝑓𝑥) ↔ 𝑏 ∈ (𝑓𝑢)))
7672, 75anbi12d 747 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑏) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) ↔ (𝑢𝐴𝑏 ∈ (𝑓𝑢))))
7769, 70, 76opelopaba 5020 . . . . . . . . 9 (⟨𝑢, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ (𝑢𝐴𝑏 ∈ (𝑓𝑢)))
7868, 77syl6bb 276 . . . . . . . 8 (((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → (⟨𝑎, 𝑏⟩ ∈ 𝑟 ↔ (𝑢𝐴𝑏 ∈ (𝑓𝑢))))
7978rabbidv 3220 . . . . . . 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 766 . . . . . . . 8 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → 𝐵𝑊)
81 rabexg 4844 . . . . . . . 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 6327 . . . . . 6 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑢) = {𝑏𝐵 ∣ (𝑢𝐴𝑏 ∈ (𝑓𝑢))})
8455, 83eqtr4d 2688 . . . . 5 ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑢𝐴) → (𝑓𝑢) = ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑢))
8535, 41, 84eqfnfvd 6354 . . . 4 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
86 simplrl 817 . . . . . . . 8 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
8786elpwid 4203 . . . . . . 7 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
88 xpss 5159 . . . . . . 7 (𝐴 × 𝐵) ⊆ (V × V)
8987, 88syl6ss 3648 . . . . . 6 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V))
90 df-rel 5150 . . . . . 6 (Rel 𝑟𝑟 ⊆ (V × V))
9189, 90sylibr 224 . . . . 5 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel 𝑟)
92 relopab 5280 . . . . . 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 472 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
952, 94anim12i 589 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) → (𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)))
9695anim1i 591 . . . . 5 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
97 vex 3234 . . . . . . . 8 𝑣 ∈ V
98 simpl 472 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
9998eleq1d 2715 . . . . . . . . 9 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑥𝐴𝑢𝐴))
100 simpr 476 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
10198fveq2d 6233 . . . . . . . . . 10 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑓𝑥) = (𝑓𝑢))
102100, 101eleq12d 2724 . . . . . . . . 9 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝑦 ∈ (𝑓𝑥) ↔ 𝑣 ∈ (𝑓𝑢)))
10399, 102anbi12d 747 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) ↔ (𝑢𝐴𝑣 ∈ (𝑓𝑢))))
10469, 97, 103opelopaba 5020 . . . . . . 7 (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ (𝑢𝐴𝑣 ∈ (𝑓𝑢)))
105 breq2 4689 . . . . . . . . . . . 12 (𝑏 = 𝑣 → (𝑢𝑟𝑏𝑢𝑟𝑣))
106 df-br 4686 . . . . . . . . . . . 12 (𝑢𝑟𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑟)
107105, 106syl6bb 276 . . . . . . . . . . 11 (𝑏 = 𝑣 → (𝑢𝑟𝑏 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑟))
108107elrab 3396 . . . . . . . . . 10 (𝑣 ∈ {𝑏𝐵𝑢𝑟𝑏} ↔ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))
109108anbi2i 730 . . . . . . . . 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 807 . . . . . . . . . . . 12 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
112 breq1 4688 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (𝑥𝑟𝑦𝑎𝑟𝑦))
113112rabbidv 3220 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → {𝑦𝐵𝑥𝑟𝑦} = {𝑦𝐵𝑎𝑟𝑦})
114 breq2 4689 . . . . . . . . . . . . . . 15 (𝑦 = 𝑏 → (𝑎𝑟𝑦𝑎𝑟𝑏))
115114cbvrabv 3230 . . . . . . . . . . . . . 14 {𝑦𝐵𝑎𝑟𝑦} = {𝑏𝐵𝑎𝑟𝑏}
116113, 115syl6eq 2701 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → {𝑦𝐵𝑥𝑟𝑦} = {𝑏𝐵𝑎𝑟𝑏})
117116cbvmptv 4783 . . . . . . . . . . . 12 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑎𝐴 ↦ {𝑏𝐵𝑎𝑟𝑏})
118111, 117syl6eq 2701 . . . . . . . . . . 11 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → 𝑓 = (𝑎𝐴 ↦ {𝑏𝐵𝑎𝑟𝑏}))
119 breq1 4688 . . . . . . . . . . . . 13 (𝑎 = 𝑢 → (𝑎𝑟𝑏𝑢𝑟𝑏))
120119rabbidv 3220 . . . . . . . . . . . 12 (𝑎 = 𝑢 → {𝑏𝐵𝑎𝑟𝑏} = {𝑏𝐵𝑢𝑟𝑏})
121120adantl 481 . . . . . . . . . . 11 (((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) ∧ 𝑎 = 𝑢) → {𝑏𝐵𝑎𝑟𝑏} = {𝑏𝐵𝑢𝑟𝑏})
122 simpr 476 . . . . . . . . . . 11 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → 𝑢𝐴)
123 rabexg 4844 . . . . . . . . . . . 12 (𝐵𝑊 → {𝑏𝐵𝑢𝑟𝑏} ∈ V)
124123ad3antrrr 766 . . . . . . . . . . 11 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → {𝑏𝐵𝑢𝑟𝑏} ∈ V)
125118, 121, 122, 124fvmptd 6327 . . . . . . . . . 10 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → (𝑓𝑢) = {𝑏𝐵𝑢𝑟𝑏})
126125eleq2d 2716 . . . . . . . . 9 ((((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑢𝐴) → (𝑣 ∈ (𝑓𝑢) ↔ 𝑣 ∈ {𝑏𝐵𝑢𝑟𝑏}))
127126pm5.32da 674 . . . . . . . 8 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ (𝑢𝐴𝑣 ∈ {𝑏𝐵𝑢𝑟𝑏})))
128 simplr 807 . . . . . . . . . 10 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
129128elpwid 4203 . . . . . . . . 9 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
13069, 97opeldm 5360 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ ∈ 𝑟𝑢 ∈ dom 𝑟)
131 dmss 5355 . . . . . . . . . . . . . 14 (𝑟 ⊆ (𝐴 × 𝐵) → dom 𝑟 ⊆ dom (𝐴 × 𝐵))
132 dmxpss 5600 . . . . . . . . . . . . . 14 dom (𝐴 × 𝐵) ⊆ 𝐴
133131, 132syl6ss 3648 . . . . . . . . . . . . 13 (𝑟 ⊆ (𝐴 × 𝐵) → dom 𝑟𝐴)
134133sseld 3635 . . . . . . . . . . . 12 (𝑟 ⊆ (𝐴 × 𝐵) → (𝑢 ∈ dom 𝑟𝑢𝐴))
135130, 134syl5 34 . . . . . . . . . . 11 (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟𝑢𝐴))
136135pm4.71rd 668 . . . . . . . . . 10 (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑢𝐴 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟)))
13769, 97opelrn 5389 . . . . . . . . . . . . 13 (⟨𝑢, 𝑣⟩ ∈ 𝑟𝑣 ∈ ran 𝑟)
138 rnss 5386 . . . . . . . . . . . . . . 15 (𝑟 ⊆ (𝐴 × 𝐵) → ran 𝑟 ⊆ ran (𝐴 × 𝐵))
139 rnxpss 5601 . . . . . . . . . . . . . . 15 ran (𝐴 × 𝐵) ⊆ 𝐵
140138, 139syl6ss 3648 . . . . . . . . . . . . . 14 (𝑟 ⊆ (𝐴 × 𝐵) → ran 𝑟𝐵)
141140sseld 3635 . . . . . . . . . . . . 13 (𝑟 ⊆ (𝐴 × 𝐵) → (𝑣 ∈ ran 𝑟𝑣𝐵))
142137, 141syl5 34 . . . . . . . . . . . 12 (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟𝑣𝐵))
143142pm4.71rd 668 . . . . . . . . . . 11 (𝑟 ⊆ (𝐴 × 𝐵) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟)))
144143anbi2d 740 . . . . . . . . . 10 (𝑟 ⊆ (𝐴 × 𝐵) → ((𝑢𝐴 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟) ↔ (𝑢𝐴 ∧ (𝑣𝐵 ∧ ⟨𝑢, 𝑣⟩ ∈ 𝑟))))
145136, 144bitrd 268 . . . . . . . . 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 300 . . . . . . 7 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑟))
148104, 147syl5rbb 273 . . . . . 6 (((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑢, 𝑣⟩ ∈ 𝑟 ↔ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
149148eqrelrdv2 5253 . . . . 5 (((Rel 𝑟 ∧ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ ((𝐵𝑊𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
15091, 93, 96, 149syl21anc 1365 . . . 4 (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
15185, 150impbida 895 . . 3 ((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))) → (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
1521, 14, 32, 151f1ocnv2d 6928 . 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 38612 . . . 4 (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
156153, 155syl5eq 2697 . . 3 (𝜑𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
157 f1oeq1 6165 . . . 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 5328 . . . . 5 (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
159158eqeq1d 2653 . . . 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 747 . . 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 247 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 196   ∧ wa 383   = wceq 1523  ⊤wtru 1524   ∈ wcel 2030  ∀wral 2941  {crab 2945  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  𝒫 cpw 4191  ⟨cop 4216   class class class wbr 4685  {copab 4745   ↦ cmpt 4762   × cxp 5141  ◡ccnv 5142  dom cdm 5143  ran crn 5144  Rel wrel 5148   Fn wfn 5921  ⟶wf 5922  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692   ↑𝑚 cmap 7899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901 This theorem is referenced by:  rfovcnvd  38616  rfovf1od  38617
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