Theorem ustfn 24145

Description: The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)

Assertion

Ref	Expression
ustfn	⊢ UnifOn Fn V

Proof of Theorem ustfn

Dummy variables 𝑣 𝑢 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		velpw 4585	. . . . 5 ⊢ (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
2	1	abbii 2803	. . . 4 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
3		abid2 2873	. . . . 5 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
4		vex 3468	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4, 4	xpex 7752	. . . . . . 7 ⊢ (𝑥 × 𝑥) ∈ V
6	5	pwex 5355	. . . . . 6 ⊢ 𝒫 (𝑥 × 𝑥) ∈ V
7	6	pwex 5355	. . . . 5 ⊢ 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
8	3, 7	eqeltri 2831	. . . 4 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
9	2, 8	eqeltrri 2832	. . 3 ⊢ {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)} ∈ V
10		simp1 1136	. . . 4 ⊢ ((𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣))) → 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
11	10	ss2abi 4047	. . 3 ⊢ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))} ⊆ {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
12	9, 11	ssexi 5297	. 2 ⊢ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))} ∈ V
13		df-ust 24144	. 2 ⊢ UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))})
14	12, 13	fnmpti 6686	1 ⊢ UnifOn Fn V

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2109  {cab 2714  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580   I cid 5552   × cxp 5657  ^◡ccnv 5658   ↾ cres 5661   ∘ ccom 5663   Fn wfn 6531  UnifOncust 24143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-fun 6538  df-fn 6539  df-ust 24144

This theorem is referenced by:  ustn0  24164  ustbas  24171