Theorem dftpos5 49685

Description: Alternate definition of tpos. (Contributed by Zhi Wang, 6-Oct-2025.)

Assertion

Ref	Expression
dftpos5	⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))

Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos5

Step	Hyp	Ref	Expression
1		df-tpos 8224	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2		mptun 6685	. . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
3		0ex 5272	. . . . . 6 ⊢ ∅ ∈ V
4		sneq 4601	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → {𝑥} = {∅})
5	4	cnveqd 5863	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ◡{𝑥} = ◡{∅})
6	5	unieqd 4887	. . . . . . . 8 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∪ ◡{∅})
7		cnvsn0 6213	. . . . . . . . . 10 ⊢ ◡{∅} = ∅
8	7	unieqi 4886	. . . . . . . . 9 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4903	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2788	. . . . . . . 8 ⊢ ∪ ◡{∅} = ∅
11	6, 10	eqtrdi 2816	. . . . . . 7 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∅)
12	11	fmptsng 7170	. . . . . 6 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
13	3, 3, 12	mp2an 705	. . . . 5 ⊢ {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥})
14	13	uneq2i 4119	. . . 4 ⊢ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
15	2, 14	eqtr4i 2791	. . 3 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})
16	15	coeq2i 5848	. 2 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
17	1, 16	eqtri 2788	1 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∪ cun 3904  ∅c0 4286  {csn 4591  ⟨cop 4597  ∪ cuni 4874   ↦ cmpt 5194  ^◡ccnv 5662  dom cdm 5663   ∘ ccom 5667  tpos ctpos 8223

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-tpos 8224

This theorem is used by:  dftpos6  49686