Theorem dftpos6 49036

Description: Alternate definition of tpos. The second half of the right hand side could apply ressn 6240 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 49035	. 2 ⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
2		coundi 6202	. 2 ⊢ (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩}))
3		0ex 5249	. . . 4 ⊢ ∅ ∈ V
4	3, 3	cosni 48996	. . 3 ⊢ (𝐹 ∘ {⟨∅, ∅⟩}) = ({∅} × (𝐹 “ {∅}))
5	4	uneq2i 4114	. 2 ⊢ ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ (𝐹 ∘ {⟨∅, ∅⟩})) = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))
6	1, 2, 5	3eqtri 2760	1 ⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))

Syntax hints:   = wceq 1541   ∪ cun 3896  ∅c0 4282  {csn 4577  ⟨cop 4583  ∪ cuni 4860   ↦ cmpt 5176   × cxp 5619  ^◡ccnv 5620  dom cdm 5621   “ cima 5624   ∘ ccom 5625  tpos ctpos 8164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-tpos 8165

This theorem is referenced by:  tposres3  49042