Theorem dftpos5 48850

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 8207	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2		mptun 6666	. . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
3		0ex 5264	. . . . . 6 ⊢ ∅ ∈ V
4		sneq 4601	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → {𝑥} = {∅})
5	4	cnveqd 5841	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ◡{𝑥} = ◡{∅})
6	5	unieqd 4886	. . . . . . . 8 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∪ ◡{∅})
7		cnvsn0 6185	. . . . . . . . . 10 ⊢ ◡{∅} = ∅
8	7	unieqi 4885	. . . . . . . . 9 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4901	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2753	. . . . . . . 8 ⊢ ∪ ◡{∅} = ∅
11	6, 10	eqtrdi 2781	. . . . . . 7 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∅)
12	11	fmptsng 7144	. . . . . 6 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
13	3, 3, 12	mp2an 692	. . . . 5 ⊢ {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥})
14	13	uneq2i 4130	. . . 4 ⊢ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
15	2, 14	eqtr4i 2756	. . 3 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})
16	15	coeq2i 5826	. 2 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
17	1, 16	eqtri 2753	1 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∪ cun 3914  ∅c0 4298  {csn 4591  ⟨cop 4597  ∪ cuni 4873   ↦ cmpt 5190  ^◡ccnv 5639  dom cdm 5640   ∘ ccom 5644  tpos ctpos 8206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-tpos 8207

This theorem is referenced by:  dftpos6  48851