Theorem dftpos5 49456

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 8200	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2		mptun 6662	. . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
3		0ex 5254	. . . . . 6 ⊢ ∅ ∈ V
4		sneq 4589	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → {𝑥} = {∅})
5	4	cnveqd 5843	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ◡{𝑥} = ◡{∅})
6	5	unieqd 4875	. . . . . . . 8 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∪ ◡{∅})
7		cnvsn0 6192	. . . . . . . . . 10 ⊢ ◡{∅} = ∅
8	7	unieqi 4874	. . . . . . . . 9 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4891	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2784	. . . . . . . 8 ⊢ ∪ ◡{∅} = ∅
11	6, 10	eqtrdi 2812	. . . . . . 7 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∅)
12	11	fmptsng 7147	. . . . . 6 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
13	3, 3, 12	mp2an 702	. . . . 5 ⊢ {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥})
14	13	uneq2i 4116	. . . 4 ⊢ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
15	2, 14	eqtr4i 2787	. . 3 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})
16	15	coeq2i 5828	. 2 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
17	1, 16	eqtri 2784	1 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))

Syntax hints:   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∪ cun 3900  ∅c0 4283  {csn 4579  ⟨cop 4585  ∪ cuni 4862   ↦ cmpt 5178  ^◡ccnv 5642  dom cdm 5643   ∘ ccom 5647  tpos ctpos 8199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-tpos 8200

This theorem is referenced by:  dftpos6  49457