Theorem dftpos5 49652

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 8218	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2		mptun 6681	. . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
3		0ex 5270	. . . . . 6 ⊢ ∅ ∈ V
4		sneq 4599	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → {𝑥} = {∅})
5	4	cnveqd 5861	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ◡{𝑥} = ◡{∅})
6	5	unieqd 4885	. . . . . . . 8 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∪ ◡{∅})
7		cnvsn0 6211	. . . . . . . . . 10 ⊢ ◡{∅} = ∅
8	7	unieqi 4884	. . . . . . . . 9 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4901	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2786	. . . . . . . 8 ⊢ ∪ ◡{∅} = ∅
11	6, 10	eqtrdi 2814	. . . . . . 7 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∅)
12	11	fmptsng 7166	. . . . . 6 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
13	3, 3, 12	mp2an 704	. . . . 5 ⊢ {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥})
14	13	uneq2i 4119	. . . 4 ⊢ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
15	2, 14	eqtr4i 2789	. . 3 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})
16	15	coeq2i 5846	. 2 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
17	1, 16	eqtri 2786	1 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))

Syntax hints:   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∪ cun 3903  ∅c0 4286  {csn 4589  ⟨cop 4595  ∪ cuni 4872   ↦ cmpt 5192  ^◡ccnv 5660  dom cdm 5661   ∘ ccom 5665  tpos ctpos 8217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-tpos 8218

This theorem is referenced by:  dftpos6  49653