Theorem fsovfvd 44777

Description: Value of the operator, (𝐴𝑂𝐵), 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, 𝐴 and 𝐵, when applied to function 𝐹. (Contributed by RP, 25-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)
fsovfvd.f	⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑m 𝐴))

Assertion

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

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

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐵(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 fsovfvd

Step	Hyp	Ref	Expression
1		fsovfvd.g	. . 3 ⊢ 𝐺 = (𝐴𝑂𝐵)
2		fsovd.fs	. . . 4 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
3		fsovd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		fsovd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5	2, 3, 4	fsovd 44775	. . 3 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
6	1, 5	eqtrid 2813	. 2 ⊢ (𝜑 → 𝐺 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
7		fveq1 6887	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
8	7	eleq2d 2852	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑦 ∈ (𝐹‘𝑥)))
9	8	rabbidv 3426	. . . 4 ⊢ (𝑓 = 𝐹 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)})
10	9	mpteq2dv 5210	. . 3 ⊢ (𝑓 = 𝐹 → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
11	10	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
12		fsovfvd.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑m 𝐴))
13	4	mptexd 7229	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}) ∈ V)
14	6, 11, 12, 13	fvmptd 7004	1 ⊢ (𝜑 → (𝐺‘𝐹) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {crab 3419  Vcvv 3458  𝒫 cpw 4567   ↦ cmpt 5197  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425   ↑_m cmap 8833

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428

This theorem is used by:  fsovfvfvd  44778  fsovcnvfvd  44782