Theorem rfovfvfvd 43986

Description: Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, relation 𝑅, and left element 𝑋. (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)
rfovfvfvd.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
rfovfvfvd.g	⊢ 𝐺 = (𝐹‘𝑅)

Assertion

Ref	Expression
rfovfvfvd	⊢ (𝜑 → (𝐺‘𝑋) = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦})

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

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

Proof of Theorem rfovfvfvd

Step	Hyp	Ref	Expression
1		rfovfvfvd.g	. . 3 ⊢ 𝐺 = (𝐹‘𝑅)
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		rfovfvd.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
6		rfovfvd.f	. . . 4 ⊢ 𝐹 = (𝐴𝑂𝐵)
7	2, 3, 4, 5, 6	rfovfvd 43985	. . 3 ⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))
8	1, 7	eqtrid 2776	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))
9		breq1 5095	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦))
10	9	rabbidv 3402	. . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦})
11	10	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦})
12		rfovfvfvd.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
13		rabexg 5276	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦} ∈ V)
14	4, 13	syl 17	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦} ∈ V)
15	8, 11, 12, 14	fvmptd 6937	1 ⊢ (𝜑 → (𝐺‘𝑋) = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3394  Vcvv 3436  𝒫 cpw 4551   class class class wbr 5092   ↦ cmpt 5173   × cxp 5617  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354

This theorem is referenced by: (None)