Theorem dftpos5 48835

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 8182	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2		mptun 6646	. . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
3		0ex 5257	. . . . . 6 ⊢ ∅ ∈ V
4		sneq 4595	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → {𝑥} = {∅})
5	4	cnveqd 5829	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ◡{𝑥} = ◡{∅})
6	5	unieqd 4880	. . . . . . . 8 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∪ ◡{∅})
7		cnvsn0 6171	. . . . . . . . . 10 ⊢ ◡{∅} = ∅
8	7	unieqi 4879	. . . . . . . . 9 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4895	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2752	. . . . . . . 8 ⊢ ∪ ◡{∅} = ∅
11	6, 10	eqtrdi 2780	. . . . . . 7 ⊢ (𝑥 = ∅ → ∪ ◡{𝑥} = ∅)
12	11	fmptsng 7124	. . . . . 6 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
13	3, 3, 12	mp2an 692	. . . . 5 ⊢ {⟨∅, ∅⟩} = (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥})
14	13	uneq2i 4124	. . . 4 ⊢ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ (𝑥 ∈ {∅} ↦ ∪ ◡{𝑥}))
15	2, 14	eqtr4i 2755	. . 3 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})
16	15	coeq2i 5814	. 2 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
17	1, 16	eqtri 2752	1 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∪ cun 3909  ∅c0 4292  {csn 4585  ⟨cop 4591  ∪ cuni 4867   ↦ cmpt 5183  ^◡ccnv 5630  dom cdm 5631   ∘ ccom 5635  tpos ctpos 8181

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-tpos 8182

This theorem is referenced by:  dftpos6  48836