Theorem dftpos6 48851

Description: Alternate definition of tpos. The second half of the right hand side could apply ressn 6260 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 48850	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
2		coundi 6222	. 2 ⊢ (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩}))
3		0ex 5264	. . . 4 ⊢ ∅ ∈ V
4	3, 3	cosni 48813	. . 3 ⊢ (𝐹 ∘ {⟨∅, ∅⟩}) = ({∅} × (𝐹 “ {∅}))
5	4	uneq2i 4130	. 2 ⊢ ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))
6	1, 2, 5	3eqtri 2757	1 ⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))

Syntax hints:   = wceq 1540   ∪ cun 3914  ∅c0 4298  {csn 4591  ⟨cop 4597  ∪ cuni 4873   ↦ cmpt 5190   × cxp 5638  ^◡ccnv 5639  dom cdm 5640   “ cima 5643   ∘ 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-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  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-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-tpos 8207

This theorem is referenced by:  tposres3  48857