Theorem dmtposss 48907

Description: The domain of tpos 𝐹 is a subset. (Contributed by Zhi Wang, 6-Oct-2025.)

Assertion

Ref	Expression
dmtposss	⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})

Proof of Theorem dmtposss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tpos 8151	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2	1	dmeqi 5839	. 2 ⊢ dom tpos 𝐹 = dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
3		dmcoss 5909	. . 3 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
4		eqid 2731	. . . . 5 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
5	4	dmmptss 6183	. . . 4 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅})
6		relcnv 6048	. . . . . 6 ⊢ Rel ◡dom 𝐹
7		df-rel 5618	. . . . . 6 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
8	6, 7	mpbi 230	. . . . 5 ⊢ ◡dom 𝐹 ⊆ (V × V)
9		unss1 4130	. . . . 5 ⊢ (◡dom 𝐹 ⊆ (V × V) → (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
10	8, 9	ax-mp 5	. . . 4 ⊢ (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅})
11	5, 10	sstri 3939	. . 3 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ ((V × V) ∪ {∅})
12	3, 11	sstri 3939	. 2 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ ((V × V) ∪ {∅})
13	2, 12	eqsstri 3976	1 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})

Syntax hints:  Vcvv 3436   ∪ cun 3895   ⊆ wss 3897  ∅c0 4278  {csn 4571  ∪ cuni 4854   ↦ cmpt 5167   × cxp 5609  ^◡ccnv 5610  dom cdm 5611   ∘ ccom 5615  Rel wrel 5616  tpos ctpos 8150

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-tpos 8151

This theorem is referenced by:  tposresg  48909