Theorem fsovcnvfvd 44005

Description: The value of the converse of (𝐴𝑂𝐵), where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, evaluated at function 𝐹. (Contributed by RP, 27-Apr-2021.)

Hypotheses

Ref	Expression
fsovd.fs	⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
fsovd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fsovd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fsovfvd.g	⊢ 𝐺 = (𝐴𝑂𝐵)
fsovcnvfvd.f	⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐴 ↑m 𝐵))

Assertion

Ref	Expression
fsovcnvfvd	⊢ (𝜑 → (◡𝐺‘𝐹) = (𝑦 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ 𝑦 ∈ (𝐹‘𝑥)}))

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

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

Proof of Theorem fsovcnvfvd

Step	Hyp	Ref	Expression
1		fsovd.fs	. . . 4 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
2		fsovd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fsovd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4		fsovfvd.g	. . . 4 ⊢ 𝐺 = (𝐴𝑂𝐵)
5		eqid 2735	. . . 4 ⊢ (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
6	1, 2, 3, 4, 5	fsovcnvd 44004	. . 3 ⊢ (𝜑 → ◡𝐺 = (𝐵𝑂𝐴))
7	6	fveq1d 6909	. 2 ⊢ (𝜑 → (◡𝐺‘𝐹) = ((𝐵𝑂𝐴)‘𝐹))
8		fsovcnvfvd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐴 ↑m 𝐵))
9	1, 3, 2, 5, 8	fsovfvd 44000	. 2 ⊢ (𝜑 → ((𝐵𝑂𝐴)‘𝐹) = (𝑦 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
10	7, 9	eqtrd 2775	1 ⊢ (𝜑 → (◡𝐺‘𝐹) = (𝑦 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ 𝑦 ∈ (𝐹‘𝑥)}))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478  𝒫 cpw 4605   ↦ cmpt 5231  ^◡ccnv 5688  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433   ↑_m cmap 8865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867

This theorem is referenced by: (None)