Theorem neicvgf1o 38932

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 38930	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
6		pwexg 4999	. . . . 5 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
8		neicvg.f	. . . 4 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
9	1, 7, 5, 8	fsovf1od 38830	. . 3 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵))
10		neicvg.p	. . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
11	10, 2, 5	dssmapf1od 38835	. . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵))
12		neicvg.g	. . . . 5 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)
13	1, 5, 7, 12	fsovf1od 38830	. . . 4 ⊢ (𝜑 → 𝐺:(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵))
14		f1oco 6321	. . . 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))
15	11, 13, 14	syl2anc 696	. . 3 ⊢ (𝜑 → (𝐷 ∘ 𝐺):(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵))
16		f1oco 6321	. . 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))
17	9, 15, 16	syl2anc 696	. 2 ⊢ (𝜑 → (𝐹 ∘ (𝐷 ∘ 𝐺)):(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵))
18		f1oeq1 6289	. . 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)))
19	3, 18	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))
20	17, 19	sylibr 224	1 ⊢ (𝜑 → 𝐻:(𝒫 𝒫 𝐵 ↑𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632   ∈ wcel 2139  {crab 3054  Vcvv 3340   ∖ cdif 3712  𝒫 cpw 4302   class class class wbr 4804   ↦ cmpt 4881   ∘ ccom 5270  –1-1-onto→wf1o 6048  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816   ↑_𝑚 cmap 8025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027

This theorem is referenced by: (None)