Theorem fsovf1od 44458

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 44454	. . 3 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)⟶(𝒫 𝐴 ↑m 𝐵))
6	5	ffnd 6661	. 2 ⊢ (𝜑 → 𝐺 Fn (𝒫 𝐵 ↑m 𝐴))
7		eqid 2737	. . . . 5 ⊢ (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
8	1, 3, 2, 7	fsovfd 44454	. . . 4 ⊢ (𝜑 → (𝐵𝑂𝐴):(𝒫 𝐴 ↑m 𝐵)⟶(𝒫 𝐵 ↑m 𝐴))
9	8	ffnd 6661	. . 3 ⊢ (𝜑 → (𝐵𝑂𝐴) Fn (𝒫 𝐴 ↑m 𝐵))
10	1, 2, 3, 4, 7	fsovcnvd 44456	. . . 4 ⊢ (𝜑 → ◡𝐺 = (𝐵𝑂𝐴))
11	10	fneq1d 6583	. . 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 6780	. 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 584	1 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→(𝒫 𝐴 ↑m 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3390  Vcvv 3430  𝒫 cpw 4542   ↦ cmpt 5167  ^◡ccnv 5621   Fn wfn 6485  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7358   ∈ cmpo 7360   ↑_m cmap 8764

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8766

This theorem is referenced by:  ntrneif1o  44517  clsneif1o  44546  clsneikex  44548  clsneinex  44549  neicvgf1o  44556  neicvgmex  44559  neicvgel1  44561