Theorem ussval 22063

Description: The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6185 to avoid requiring 𝑊 to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)

Hypotheses

Ref	Expression
ussval.1	⊢ 𝐵 = (Base‘𝑊)
ussval.2	⊢ 𝑈 = (UnifSet‘𝑊)

Assertion

Ref	Expression
ussval	⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)

Proof of Theorem ussval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊))
2		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3	2	sqxpeqd 5141	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
4	1, 3	oveq12d 6668	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
5		df-uss 22060	. . . 4 ⊢ UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))))
6		ovex 6678	. . . 4 ⊢ ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V
7	4, 5, 6	fvmpt 6282	. . 3 ⊢ (𝑊 ∈ V → (UnifSt‘𝑊) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
8		ussval.2	. . . 4 ⊢ 𝑈 = (UnifSet‘𝑊)
9		ussval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
10	9, 9	xpeq12i 5137	. . . 4 ⊢ (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
11	8, 10	oveq12i 6662	. . 3 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))
12	7, 11	syl6reqr 2675	. 2 ⊢ (𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
13		0rest 16090	. . 3 ⊢ (∅ ↾t (𝐵 × 𝐵)) = ∅
14		fvprc 6185	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
15	8, 14	syl5eq 2668	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
16	15	oveq1d 6665	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17		fvprc 6185	. . 3 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
18	13, 16, 17	3eqtr4a 2682	. 2 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
19	12, 18	pm2.61i 176	1 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915   × cxp 5112  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  UnifSetcunif 15951   ↾_t crest 16081  UnifStcuss 22057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-rest 16083  df-uss 22060

This theorem is referenced by:  ussid  22064  ressuss  22067