Theorem rfovd 43431

Description: Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 25-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)

Assertion

Ref	Expression
rfovd	⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑟)   𝐴(𝑦)   𝑂(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑟,𝑎,𝑏)

Proof of Theorem rfovd

Step	Hyp	Ref	Expression
1		rfovd.rf	. . 3 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))))
3		xpeq12 5703	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
4	3	pweqd 4620	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
5		simpl 482	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴)
6		rabeq 3443	. . . . . 6 ⊢ (𝑏 = 𝐵 → {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
7	6	adantl 481	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})
8	5, 7	mpteq12dv 5239	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))
9	4, 8	mpteq12dv 5239	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
10	9	adantl 481	. 2 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
11		rfovd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12	11	elexd 3492	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
13		rfovd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
14	13	elexd 3492	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
15	11, 13	xpexd 7753	. . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
16		pwexg 5378	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
17		mptexg 7233	. . 3 ⊢ (𝒫 (𝐴 × 𝐵) ∈ V → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∈ V)
18	15, 16, 17	3syl 18	. 2 ⊢ (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∈ V)
19	2, 10, 12, 14, 18	ovmpod 7573	1 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3429  Vcvv 3471  𝒫 cpw 4603   class class class wbr 5148   ↦ cmpt 5231   × cxp 5676  (class class class)co 7420   ∈ cmpo 7422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is referenced by:  rfovfvd  43432  rfovcnvf1od  43434  fsovrfovd  43439