Theorem dftpos6 49653

Description: Alternate definition of tpos. The second half of the right hand side could apply ressn 6286 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 49652	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
2		coundi 6248	. 2 ⊢ (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩}))
3		0ex 5270	. . . 4 ⊢ ∅ ∈ V
4	3, 3	cosni 49613	. . 3 ⊢ (𝐹 ∘ {⟨∅, ∅⟩}) = ({∅} × (𝐹 “ {∅}))
5	4	uneq2i 4119	. 2 ⊢ ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))
6	1, 2, 5	3eqtri 2790	1 ⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))

Syntax hints:   = wceq 1570   ∪ cun 3903  ∅c0 4286  {csn 4589  ⟨cop 4595  ∪ cuni 4872   ↦ cmpt 5192   × cxp 5659  ^◡ccnv 5660  dom cdm 5661   “ cima 5664   ∘ ccom 5665  tpos ctpos 8217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-tpos 8218

This theorem is referenced by:  tposres3  49659