Theorem dmtposss 49538

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 8221	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2	1	dmeqi 5895	. 2 ⊢ dom tpos 𝐹 = dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
3		dmcoss 5966	. . 3 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
4		eqid 2769	. . . . 5 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
5	4	dmmptss 6243	. . . 4 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅})
6		relcnv 6107	. . . . . 6 ⊢ Rel ◡dom 𝐹
7		df-rel 5669	. . . . . 6 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
8	6, 7	mpbi 233	. . . . 5 ⊢ ◡dom 𝐹 ⊆ (V × V)
9		unss1 4146	. . . . 5 ⊢ (◡dom 𝐹 ⊆ (V × V) → (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
10	8, 9	ax-mp 5	. . . 4 ⊢ (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅})
11	5, 10	sstri 3954	. . 3 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ ((V × V) ∪ {∅})
12	3, 11	sstri 3954	. 2 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ ((V × V) ∪ {∅})
13	2, 12	eqsstri 3991	1 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})

Syntax hints:  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  {csn 4594  ∪ cuni 4876   ↦ cmpt 5196   × cxp 5660  ^◡ccnv 5661  dom cdm 5662   ∘ ccom 5666  Rel wrel 5667  tpos ctpos 8220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-mpt 5197  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-tpos 8221

This theorem is referenced by:  tposresg  49540