Theorem rfovfvfvd 43965

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 43964	. . 3 ⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))
8	1, 7	eqtrid 2792	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))
9		breq1 5169	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦))
10	9	rabbidv 3451	. . 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 5355	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦} ∈ V)
14	4, 13	syl 17	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦} ∈ V)
15	8, 11, 12, 14	fvmptd 7036	1 ⊢ (𝜑 → (𝐺‘𝑋) = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488  𝒫 cpw 4622   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453

This theorem is referenced by: (None)