Theorem rfovcnvd 45148

Description: Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (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)

Assertion

Ref	Expression
rfovcnvd	⊢ (𝜑 → ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))

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

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

Proof of Theorem rfovcnvd

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	rfovcnvf1od 45147	. 2 ⊢ (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})))
6	5	simprd 501	1 ⊢ (𝜑 → ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {crab 3416  Vcvv 3454  𝒫 cpw 4560   class class class wbr 5106  {copab 5170   ↦ cmpt 5189   × cxp 5653  ^◡ccnv 5654  –1-1-onto→wf1o 6546  ‘cfv 6547  (class class class)co 7430   ∈ cmpo 7432   ↑_m cmap 8861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8011  df-2nd 8012  df-map 8863

This theorem is used by:  rfovcnvfvd  45150  fsovrfovd  45152