Theorem dftpos6 50124

Description: Alternate definition of tpos. The second half of the right hand side could apply ressn 6291 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 50123	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
2		coundi 6251	. 2 ⊢ (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩}))
3		0ex 5260	. . . 4 ⊢ ∅ ∈ V
4	3, 3	cosni 50082	. . 3 ⊢ (𝐹 ∘ {⟨∅, ∅⟩}) = ({∅} × (𝐹 “ {∅}))
5	4	uneq2i 4163	. 2 ⊢ ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))
6	1, 2, 5	3eqtri 2787	1 ⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))

This proof depends on syntax axioms:   = wceq 1569   ∪ cun 4148  ∅c0 4277  {csn 4584  ⟨cop 4834  ∪ cuni 4866   ↦ cmpt 5185   × cxp 5649  ^◡ccnv 5650  dom cdm 5651   “ cima 5654   ∘ ccom 5655  tpos ctpos 8250

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-tpos 8251

This theorem is used by:  tposres3  50130