Theorem rfovcnvfvd 44458

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 44456	. 2 ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑔‘𝑥))}))
6		fveq1 6833	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
7	6	eleq2d 2826	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦 ∈ (𝑔‘𝑥) ↔ 𝑦 ∈ (𝐺‘𝑥)))
8	7	anbi2d 636	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑔‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))))
9	8	opabbidv 5145	. . 3 ⊢ (𝑔 = 𝐺 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑔‘𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))})
10	9	adantl 482	. 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 776	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))) → 𝑥 ∈ 𝐴)
13		elmapi 8793	. . . . . . . 8 ⊢ (𝐺 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝐺:𝐴⟶𝒫 𝐵)
14	13	ffvelcdmda 7032	. . . . . . 7 ⊢ ((𝐺 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝒫 𝐵)
15	11, 14	sylan 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝒫 𝐵)
16	15	elpwid 4545	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ⊆ 𝐵)
17	16	sseld 3921	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝐺‘𝑥) → 𝑦 ∈ 𝐵))
18	17	impr 455	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))) → 𝑦 ∈ 𝐵)
19	2, 3, 12, 18	opabex2 8006	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))} ∈ V)
20	5, 10, 11, 19	fvmptd 6950	1 ⊢ (𝜑 → (◡𝐹‘𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432  𝒫 cpw 4536   class class class wbr 5079  {copab 5141   ↦ cmpt 5160   × cxp 5623  ^◡ccnv 5624  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365   ↑_m cmap 8770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-map 8772

This theorem is referenced by: (None)