Theorem fsovd 38302

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. (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)

Assertion

Ref	Expression
fsovd	⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))

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

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

Proof of Theorem fsovd

Step	Hyp	Ref	Expression
1		fsovd.fs	. . 3 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}))))
3		pweq 4161	. . . . . 6 ⊢ (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
4	3	adantl 482	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝒫 𝑏 = 𝒫 𝐵)
5		simpl 473	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴)
6	4, 5	oveq12d 6668	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝒫 𝑏 ↑𝑚 𝑎) = (𝒫 𝐵 ↑𝑚 𝐴))
7		simpr 477	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
8		rabeq 3192	. . . . . 6 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})
9	8	adantr 481	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})
10	7, 9	mpteq12dv 4733	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))
11	6, 10	mpteq12dv 4733	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
12	11	adantl 482	. 2 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
13		fsovd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
14	13	elexd 3214	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
15		fsovd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
16	15	elexd 3214	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
17		ovex 6678	. . . 4 ⊢ (𝒫 𝐵 ↑𝑚 𝐴) ∈ V
18	17	mptex 6486	. . 3 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})) ∈ V
19	18	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})) ∈ V)
20	2, 12, 14, 16, 19	ovmpt2d 6788	1 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200  𝒫 cpw 4158   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  fsovrfovd  38303  fsovfvd  38304  fsovfd  38306  fsovcnvlem  38307