Theorem rfovcnvfvd 37321

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 37319	. 2 ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑔‘𝑥))}))
6		fveq1 6102	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
7	6	eleq2d 2673	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦 ∈ (𝑔‘𝑥) ↔ 𝑦 ∈ (𝐺‘𝑥)))
8	7	anbi2d 736	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑔‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))))
9	8	opabbidv 4648	. . 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 790	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))) → 𝑥 ∈ 𝐴)
13		elmapi 7765	. . . . . . . 8 ⊢ (𝐺 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → 𝐺:𝐴⟶𝒫 𝐵)
14	13	ffvelrnda 6267	. . . . . . 7 ⊢ ((𝐺 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝒫 𝐵)
15	11, 14	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝒫 𝐵)
16	15	elpwid 4118	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ⊆ 𝐵)
17	16	sseld 3567	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝐺‘𝑥) → 𝑦 ∈ 𝐵))
18	17	impr 647	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))) → 𝑦 ∈ 𝐵)
19	2, 3, 12, 18	opabex2 6997	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))} ∈ V)
20	5, 10, 11, 19	fvmptd 6197	1 ⊢ (𝜑 → (◡𝐹‘𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  𝒫 cpw 4108   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746

This theorem is referenced by: (None)