Theorem neicvgf1o 44297

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 44295	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
6	5	pwexd 5322	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
7		neicvg.f	. . . 4 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
8	1, 6, 5, 7	fsovf1od 44199	. . 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 44204	. . . 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 44199	. . . 4 ⊢ (𝜑 → 𝐺:(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
13		f1oco 6795	. . . 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 6795	. . 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 6760	. . 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 1541   ∈ wcel 2113  {crab 3397  Vcvv 3438   ∖ cdif 3896  𝒫 cpw 4552   class class class wbr 5096   ↦ cmpt 5177   ∘ ccom 5626  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358   ↑_m cmap 8761

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 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  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 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763

This theorem is referenced by: (None)