Theorem dmtposss 49163

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 8170	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2	1	dmeqi 5854	. 2 ⊢ dom tpos 𝐹 = dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
3		dmcoss 5925	. . 3 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
4		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
5	4	dmmptss 6200	. . . 4 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅})
6		relcnv 6064	. . . . . 6 ⊢ Rel ◡dom 𝐹
7		df-rel 5632	. . . . . 6 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
8	6, 7	mpbi 230	. . . . 5 ⊢ ◡dom 𝐹 ⊆ (V × V)
9		unss1 4138	. . . . 5 ⊢ (◡dom 𝐹 ⊆ (V × V) → (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
10	8, 9	ax-mp 5	. . . 4 ⊢ (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅})
11	5, 10	sstri 3944	. . 3 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ ((V × V) ∪ {∅})
12	3, 11	sstri 3944	. 2 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ ((V × V) ∪ {∅})
13	2, 12	eqsstri 3981	1 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})

Syntax hints:  Vcvv 3441   ∪ cun 3900   ⊆ wss 3902  ∅c0 4286  {csn 4581  ∪ cuni 4864   ↦ cmpt 5180   × cxp 5623  ^◡ccnv 5624  dom cdm 5625   ∘ ccom 5629  Rel wrel 5630  tpos ctpos 8169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-mpt 5181  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-tpos 8170

This theorem is referenced by:  tposresg  49165