Theorem neicvgf1o 44103

Description: If neighborhood and convergent functions are related by operator 𝐻, it is a one-to-one onto relation. (Contributed by RP, 11-Jun-2021.)

Hypotheses

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

Assertion

Ref	Expression
neicvgf1o	⊢ (𝜑 → 𝐻:(𝒫 𝒫 𝐵 ↑m 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)   𝐺(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑀(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 neicvgf1o

Step	Hyp	Ref	Expression
1		neicvg.o	. . . 4 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		neicvg.d	. . . . . 6 ⊢ 𝐷 = (𝑃‘𝐵)
3		neicvg.h	. . . . . 6 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
4		neicvg.r	. . . . . 6 ⊢ (𝜑 → 𝑁𝐻𝑀)
5	2, 3, 4	neicvgbex 44101	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
6	5	pwexd 5384	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
7		neicvg.f	. . . 4 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
8	1, 6, 5, 7	fsovf1od 44005	. . 3 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
9		neicvg.p	. . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
10	9, 2, 5	dssmapf1od 44010	. . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
11		neicvg.g	. . . . 5 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)
12	1, 5, 6, 11	fsovf1od 44005	. . . 4 ⊢ (𝜑 → 𝐺:(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
13		f1oco 6871	. . . 4 ⊢ ((𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐺:(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐷 ∘ 𝐺):(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
14	10, 12, 13	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐷 ∘ 𝐺):(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
15		f1oco 6871	. . 3 ⊢ ((𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ (𝐷 ∘ 𝐺):(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐹 ∘ (𝐷 ∘ 𝐺)):(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
16	8, 14, 15	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹 ∘ (𝐷 ∘ 𝐺)):(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
17		f1oeq1 6836	. . 3 ⊢ (𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) → (𝐻:(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ↔ (𝐹 ∘ (𝐷 ∘ 𝐺)):(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)))
18	3, 17	ax-mp 5	. 2 ⊢ (𝐻:(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ↔ (𝐹 ∘ (𝐷 ∘ 𝐺)):(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
19	16, 18	sylibr 234	1 ⊢ (𝜑 → 𝐻:(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1536   ∈ wcel 2105  {crab 3432  Vcvv 3477   ∖ cdif 3959  𝒫 cpw 4604   class class class wbr 5147   ↦ cmpt 5230   ∘ ccom 5692  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432   ↑_m cmap 8864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866

This theorem is referenced by: (None)