Theorem dmtposss 49037

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 8165	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2	1	dmeqi 5850	. 2 ⊢ dom tpos 𝐹 = dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
3		dmcoss 5921	. . 3 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
4		eqid 2733	. . . . 5 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
5	4	dmmptss 6196	. . . 4 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅})
6		relcnv 6060	. . . . . 6 ⊢ Rel ◡dom 𝐹
7		df-rel 5628	. . . . . 6 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
8	6, 7	mpbi 230	. . . . 5 ⊢ ◡dom 𝐹 ⊆ (V × V)
9		unss1 4134	. . . . 5 ⊢ (◡dom 𝐹 ⊆ (V × V) → (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
10	8, 9	ax-mp 5	. . . 4 ⊢ (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅})
11	5, 10	sstri 3940	. . 3 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ ((V × V) ∪ {∅})
12	3, 11	sstri 3940	. 2 ⊢ dom (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ ((V × V) ∪ {∅})
13	2, 12	eqsstri 3977	1 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})

Syntax hints:  Vcvv 3437   ∪ cun 3896   ⊆ wss 3898  ∅c0 4282  {csn 4577  ∪ cuni 4860   ↦ cmpt 5176   × cxp 5619  ^◡ccnv 5620  dom cdm 5621   ∘ ccom 5625  Rel wrel 5626  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-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-br 5096  df-opab 5158  df-mpt 5177  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-tpos 8165

This theorem is referenced by:  tposresg  49039