Theorem ustfn 24576

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 4565	. . . . 5 ⊢ (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
2	1	abbii 2830	. . . 4 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
3		abid2 2901	. . . . 5 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
4		vex 3458	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4, 4	xpex 7799	. . . . . . 7 ⊢ (𝑥 × 𝑥) ∈ V
6	5	pwex 5349	. . . . . 6 ⊢ 𝒫 (𝑥 × 𝑥) ∈ V
7	6	pwex 5349	. . . . 5 ⊢ 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
8	3, 7	eqeltri 2860	. . . 4 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
9	2, 8	eqeltrri 2861	. . 3 ⊢ {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)} ∈ V
10		simp1 1154	. . . 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 3994	. . 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 5291	. 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 24575	. 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 6712	1 ⊢ UnifOn Fn V

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560   I cid 5552   × cxp 5656  ^◡ccnv 5657   ↾ cres 5660   ∘ ccom 5662   Fn wfn 6561  UnifOncust 24574

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-fun 6568  df-fn 6569  df-ust 24575

This theorem is used by:  ustn0  24595  ustbas  24601