Theorem rfovfvd 43514

Description: Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, and relation 𝑅. (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)
rfovfvd.r	⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
rfovfvd.f	⊢ 𝐹 = (𝐴𝑂𝐵)

Assertion

Ref	Expression
rfovfvd	⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))

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

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

Proof of Theorem rfovfvd

Step	Hyp	Ref	Expression
1		rfovfvd.f	. . 3 ⊢ 𝐹 = (𝐴𝑂𝐵)
2		rfovd.rf	. . . 4 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))
3		rfovd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		rfovd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5	2, 3, 4	rfovd 43513	. . 3 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
6	1, 5	eqtrid 2777	. 2 ⊢ (𝜑 → 𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))
7		breq 5150	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
8	7	rabbidv 3427	. . . 4 ⊢ (𝑟 = 𝑅 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦})
9	8	mpteq2dv 5250	. . 3 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))
10	9	adantl 480	. 2 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))
11		rfovfvd.r	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
12	3	mptexd 7234	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}) ∈ V)
13	6, 10, 11, 12	fvmptd 7009	1 ⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {crab 3419  Vcvv 3463  𝒫 cpw 4603   class class class wbr 5148   ↦ cmpt 5231   × cxp 5675  ‘cfv 6547  (class class class)co 7417   ∈ cmpo 7419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  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 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422

This theorem is referenced by:  rfovfvfvd  43515

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