Theorem fsovf1od 38829

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 38825	. . 3 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑𝑚 𝐴)⟶(𝒫 𝐴 ↑𝑚 𝐵))
6	5	ffnd 6186	. 2 ⊢ (𝜑 → 𝐺 Fn (𝒫 𝐵 ↑𝑚 𝐴))
7		eqid 2770	. . . . 5 ⊢ (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
8	1, 3, 2, 7	fsovfd 38825	. . . 4 ⊢ (𝜑 → (𝐵𝑂𝐴):(𝒫 𝐴 ↑𝑚 𝐵)⟶(𝒫 𝐵 ↑𝑚 𝐴))
9	8	ffnd 6186	. . 3 ⊢ (𝜑 → (𝐵𝑂𝐴) Fn (𝒫 𝐴 ↑𝑚 𝐵))
10	1, 2, 3, 4, 7	fsovcnvd 38827	. . . 4 ⊢ (𝜑 → ◡𝐺 = (𝐵𝑂𝐴))
11	10	fneq1d 6121	. . 3 ⊢ (𝜑 → (◡𝐺 Fn (𝒫 𝐴 ↑𝑚 𝐵) ↔ (𝐵𝑂𝐴) Fn (𝒫 𝐴 ↑𝑚 𝐵)))
12	9, 11	mpbird 247	. 2 ⊢ (𝜑 → ◡𝐺 Fn (𝒫 𝐴 ↑𝑚 𝐵))
13		dff1o4 6286	. 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 564	1 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑𝑚 𝐴)–1-1-onto→(𝒫 𝐴 ↑𝑚 𝐵))

Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2144  {crab 3064  Vcvv 3349  𝒫 cpw 4295   ↦ cmpt 4861  ^◡ccnv 5248   Fn wfn 6026  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794   ↑_𝑚 cmap 8008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010

This theorem is referenced by:  ntrneif1o  38892  clsneif1o  38921  clsneikex  38923  clsneinex  38924  neicvgf1o  38931  neicvgmex  38934  neicvgel1  38936