Theorem dftpos6 48748

Description: Alternate definition of tpos. The second half of the right hand side could apply ressn 6303 and become (𝐹 ↾ {∅}) (Contributed by Zhi Wang, 6-Oct-2025.)

Assertion

Ref	Expression
dftpos6	⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))

Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos6

Step	Hyp	Ref	Expression
1		dftpos5 48747	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
2		coundi 6265	. 2 ⊢ (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩}))
3		0ex 5305	. . . 4 ⊢ ∅ ∈ V
4	3, 3	cosni 48719	. . 3 ⊢ (𝐹 ∘ {⟨∅, ∅⟩}) = ({∅} × (𝐹 “ {∅}))
5	4	uneq2i 4164	. 2 ⊢ ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))
6	1, 2, 5	3eqtri 2768	1 ⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))

Syntax hints:   = wceq 1540   ∪ cun 3948  ∅c0 4332  {csn 4624  ⟨cop 4630  ∪ cuni 4905   ↦ cmpt 5223   × cxp 5681  ^◡ccnv 5682  dom cdm 5683   “ cima 5686   ∘ ccom 5687  tpos ctpos 8246

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-tpos 8247

This theorem is referenced by:  tposres3  48754