Theorem rfovcnvfvd 44434

Description: Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, evaluated at function 𝐺. (Contributed by RP, 27-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)
rfovcnvf1od.f	⊢ 𝐹 = (𝐴𝑂𝐵)
rfovcnvfv.g	⊢ (𝜑 → 𝐺 ∈ (𝒫 𝐵 ↑m 𝐴))

Assertion

Ref	Expression
rfovcnvfvd	⊢ (𝜑 → (◡𝐹‘𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))})

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

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

Proof of Theorem rfovcnvfvd

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rfovd.rf	. . 3 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))
2		rfovd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		rfovd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4		rfovcnvf1od.f	. . 3 ⊢ 𝐹 = (𝐴𝑂𝐵)
5	1, 2, 3, 4	rfovcnvd 44432	. 2 ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑔‘𝑥))}))
6		fveq1 6839	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
7	6	eleq2d 2822	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦 ∈ (𝑔‘𝑥) ↔ 𝑦 ∈ (𝐺‘𝑥)))
8	7	anbi2d 631	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑔‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))))
9	8	opabbidv 5151	. . 3 ⊢ (𝑔 = 𝐺 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑔‘𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))})
10	9	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑔‘𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))})
11		rfovcnvfv.g	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝒫 𝐵 ↑m 𝐴))
12		simprl 771	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))) → 𝑥 ∈ 𝐴)
13		elmapi 8796	. . . . . . . 8 ⊢ (𝐺 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝐺:𝐴⟶𝒫 𝐵)
14	13	ffvelcdmda 7036	. . . . . . 7 ⊢ ((𝐺 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝒫 𝐵)
15	11, 14	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝒫 𝐵)
16	15	elpwid 4550	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ⊆ 𝐵)
17	16	sseld 3920	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝐺‘𝑥) → 𝑦 ∈ 𝐵))
18	17	impr 454	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))) → 𝑦 ∈ 𝐵)
19	2, 3, 12, 18	opabex2 8010	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))} ∈ V)
20	5, 10, 11, 19	fvmptd 6955	1 ⊢ (𝜑 → (◡𝐹‘𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429  𝒫 cpw 4541   class class class wbr 5085  {copab 5147   ↦ cmpt 5166   × cxp 5629  ^◡ccnv 5630  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369   ↑_m cmap 8773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775

This theorem is referenced by: (None)