Theorem fsovf1od 41125

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 41121	. . 3 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)⟶(𝒫 𝐴 ↑m 𝐵))
6	5	ffnd 6504	. 2 ⊢ (𝜑 → 𝐺 Fn (𝒫 𝐵 ↑m 𝐴))
7		eqid 2758	. . . . 5 ⊢ (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
8	1, 3, 2, 7	fsovfd 41121	. . . 4 ⊢ (𝜑 → (𝐵𝑂𝐴):(𝒫 𝐴 ↑m 𝐵)⟶(𝒫 𝐵 ↑m 𝐴))
9	8	ffnd 6504	. . 3 ⊢ (𝜑 → (𝐵𝑂𝐴) Fn (𝒫 𝐴 ↑m 𝐵))
10	1, 2, 3, 4, 7	fsovcnvd 41123	. . . 4 ⊢ (𝜑 → ◡𝐺 = (𝐵𝑂𝐴))
11	10	fneq1d 6432	. . 3 ⊢ (𝜑 → (◡𝐺 Fn (𝒫 𝐴 ↑m 𝐵) ↔ (𝐵𝑂𝐴) Fn (𝒫 𝐴 ↑m 𝐵)))
12	9, 11	mpbird 260	. 2 ⊢ (𝜑 → ◡𝐺 Fn (𝒫 𝐴 ↑m 𝐵))
13		dff1o4 6615	. 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 586	1 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→(𝒫 𝐴 ↑m 𝐵))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409  𝒫 cpw 4497   ↦ cmpt 5116  ^◡ccnv 5527   Fn wfn 6335  –1-1-onto→wf1o 6339  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158   ↑_m cmap 8422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-map 8424

This theorem is referenced by:  ntrneif1o  41186  clsneif1o  41215  clsneikex  41217  clsneinex  41218  neicvgf1o  41225  neicvgmex  41228  neicvgel1  41230