Theorem fsovf1od 44530

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 44526	. . 3 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)⟶(𝒫 𝐴 ↑m 𝐵))
6	5	ffnd 6677	. 2 ⊢ (𝜑 → 𝐺 Fn (𝒫 𝐵 ↑m 𝐴))
7		eqid 2752	. . . . 5 ⊢ (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
8	1, 3, 2, 7	fsovfd 44526	. . . 4 ⊢ (𝜑 → (𝐵𝑂𝐴):(𝒫 𝐴 ↑m 𝐵)⟶(𝒫 𝐵 ↑m 𝐴))
9	8	ffnd 6677	. . 3 ⊢ (𝜑 → (𝐵𝑂𝐴) Fn (𝒫 𝐴 ↑m 𝐵))
10	1, 2, 3, 4, 7	fsovcnvd 44528	. . . 4 ⊢ (𝜑 → ◡𝐺 = (𝐵𝑂𝐴))
11	10	fneq1d 6599	. . 3 ⊢ (𝜑 → (◡𝐺 Fn (𝒫 𝐴 ↑m 𝐵) ↔ (𝐵𝑂𝐴) Fn (𝒫 𝐴 ↑m 𝐵)))
12	9, 11	mpbird 259	. 2 ⊢ (𝜑 → ◡𝐺 Fn (𝒫 𝐴 ↑m 𝐵))
13		dff1o4 6800	. 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 591	1 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→(𝒫 𝐴 ↑m 𝐵))

Syntax hints:   → wi 4   = wceq 1550   ∈ wcel 2132  {crab 3404  Vcvv 3444  𝒫 cpw 4545   ↦ cmpt 5171  ^◡ccnv 5635   Fn wfn 6501  –1-1-onto→wf1o 6505  ‘cfv 6506  (class class class)co 7381   ∈ cmpo 7383   ↑_m cmap 8792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-map 8794

This theorem is referenced by:  ntrneif1o  44589  clsneif1o  44618  clsneikex  44620  clsneinex  44621  neicvgf1o  44628  neicvgmex  44631  neicvgel1  44633