Theorem dmtposss 49458

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 8200	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2	1	dmeqi 5876	. 2 ⊢ dom tpos 𝐹 = dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
3		dmcoss 5947	. . 3 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
4		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
5	4	dmmptss 6223	. . . 4 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅})
6		relcnv 6089	. . . . . 6 ⊢ Rel ◡dom 𝐹
7		df-rel 5650	. . . . . 6 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
8	6, 7	mpbi 232	. . . . 5 ⊢ ◡dom 𝐹 ⊆ (V × V)
9		unss1 4135	. . . . 5 ⊢ (◡dom 𝐹 ⊆ (V × V) → (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
10	8, 9	ax-mp 5	. . . 4 ⊢ (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅})
11	5, 10	sstri 3943	. . 3 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ ((V × V) ∪ {∅})
12	3, 11	sstri 3943	. 2 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ ((V × V) ∪ {∅})
13	2, 12	eqsstri 3980	1 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})

Syntax hints:  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  ∅c0 4283  {csn 4579  ∪ cuni 4862   ↦ cmpt 5178   × cxp 5641  ^◡ccnv 5642  dom cdm 5643   ∘ ccom 5647  Rel wrel 5648  tpos ctpos 8199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-mpt 5179  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-tpos 8200

This theorem is referenced by:  tposresg  49460