Theorem clsneif1o 44221

Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the operator is a one-to-one, onto mapping. (Contributed by RP, 5-Jun-2021.)

Hypotheses

Ref	Expression
clsnei.o	⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
clsnei.p	⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
clsnei.d	⊢ 𝐷 = (𝑃‘𝐵)
clsnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
clsnei.h	⊢ 𝐻 = (𝐹 ∘ 𝐷)
clsnei.r	⊢ (𝜑 → 𝐾𝐻𝑁)

Assertion

Ref	Expression
clsneif1o	⊢ (𝜑 → 𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝐵,𝑛,𝑜,𝑝   𝜑,𝑖,𝑗,𝑘,𝑙   𝜑,𝑛,𝑜,𝑝

Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐾(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)

Proof of Theorem clsneif1o

Step	Hyp	Ref	Expression
1		clsnei.d	. . . 4 ⊢ 𝐷 = (𝑃‘𝐵)
2		clsnei.h	. . . 4 ⊢ 𝐻 = (𝐹 ∘ 𝐷)
3		clsnei.r	. . . 4 ⊢ (𝜑 → 𝐾𝐻𝑁)
4	1, 2, 3	clsneibex 44219	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
5		clsnei.o	. . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
6		pwexg 5318	. . . . . 6 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
7	6	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
8		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
9		eqid 2733	. . . . 5 ⊢ (𝒫 𝐵𝑂𝐵) = (𝒫 𝐵𝑂𝐵)
10	5, 7, 8, 9	fsovf1od 44133	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝒫 𝐵𝑂𝐵):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
11		clsnei.p	. . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
12		eqid 2733	. . . . 5 ⊢ (𝑃‘𝐵) = (𝑃‘𝐵)
13	11, 12, 8	dssmapf1od 44138	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝑃‘𝐵):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
14		f1oco 6791	. . . 4 ⊢ (((𝒫 𝐵𝑂𝐵):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ (𝑃‘𝐵):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) → ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
15	10, 13, 14	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
16	4, 15	mpdan 687	. 2 ⊢ (𝜑 → ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
17		clsnei.f	. . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
18	17, 1	coeq12i 5807	. . . 4 ⊢ (𝐹 ∘ 𝐷) = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵))
19	2, 18	eqtri 2756	. . 3 ⊢ 𝐻 = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵))
20		f1oeq1 6756	. . 3 ⊢ (𝐻 = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)) → (𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ↔ ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)))
21	19, 20	ax-mp 5	. 2 ⊢ (𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ↔ ((𝒫 𝐵𝑂𝐵) ∘ (𝑃‘𝐵)):(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
22	16, 21	sylibr 234	1 ⊢ (𝜑 → 𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3396  Vcvv 3437   ∖ cdif 3895  𝒫 cpw 4549   class class class wbr 5093   ↦ cmpt 5174   ∘ ccom 5623  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354   ↑_m cmap 8756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758

This theorem is referenced by: (None)