Theorem fsovf1od 43918

Description: The value of (𝐴𝑂𝐵) is a bijection, 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. (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)

Assertion

Ref	Expression
fsovf1od	⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→(𝒫 𝐴 ↑m 𝐵))

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

Allowed substitution hints:   𝜑(𝑥)   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 fsovf1od

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	1, 2, 3, 4	fsovfd 43914	. . 3 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)⟶(𝒫 𝐴 ↑m 𝐵))
6	5	ffnd 6747	. 2 ⊢ (𝜑 → 𝐺 Fn (𝒫 𝐵 ↑m 𝐴))
7		eqid 2734	. . . . 5 ⊢ (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
8	1, 3, 2, 7	fsovfd 43914	. . . 4 ⊢ (𝜑 → (𝐵𝑂𝐴):(𝒫 𝐴 ↑m 𝐵)⟶(𝒫 𝐵 ↑m 𝐴))
9	8	ffnd 6747	. . 3 ⊢ (𝜑 → (𝐵𝑂𝐴) Fn (𝒫 𝐴 ↑m 𝐵))
10	1, 2, 3, 4, 7	fsovcnvd 43916	. . . 4 ⊢ (𝜑 → ◡𝐺 = (𝐵𝑂𝐴))
11	10	fneq1d 6671	. . 3 ⊢ (𝜑 → (◡𝐺 Fn (𝒫 𝐴 ↑m 𝐵) ↔ (𝐵𝑂𝐴) Fn (𝒫 𝐴 ↑m 𝐵)))
12	9, 11	mpbird 257	. 2 ⊢ (𝜑 → ◡𝐺 Fn (𝒫 𝐴 ↑m 𝐵))
13		dff1o4 6869	. 2 ⊢ (𝐺:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→(𝒫 𝐴 ↑m 𝐵) ↔ (𝐺 Fn (𝒫 𝐵 ↑m 𝐴) ∧ ◡𝐺 Fn (𝒫 𝐴 ↑m 𝐵)))
14	6, 12, 13	sylanbrc 582	1 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→(𝒫 𝐴 ↑m 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2103  {crab 3438  Vcvv 3482  𝒫 cpw 4622   ↦ cmpt 5252  ^◡ccnv 5698   Fn wfn 6567  –1-1-onto→wf1o 6571  ‘cfv 6572  (class class class)co 7445   ∈ cmpo 7447   ↑_m cmap 8880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-1st 8026  df-2nd 8027  df-map 8882

This theorem is referenced by:  ntrneif1o  43977  clsneif1o  44006  clsneikex  44008  clsneinex  44009  neicvgf1o  44016  neicvgmex  44019  neicvgel1  44021