Theorem ussval 24215

Description: The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6834 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		ussval.2	. . . 4 ⊢ 𝑈 = (UnifSet‘𝑊)
2		ussval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3	2, 2	xpeq12i 5660	. . . 4 ⊢ (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
4	1, 3	oveq12i 7380	. . 3 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))
5		fveq2 6842	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊))
6		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7	6	sqxpeqd 5664	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
8	5, 7	oveq12d 7386	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
9		df-uss 24212	. . . 4 ⊢ UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))))
10		ovex 7401	. . . 4 ⊢ ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V
11	8, 9, 10	fvmpt 6949	. . 3 ⊢ (𝑊 ∈ V → (UnifSt‘𝑊) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
12	4, 11	eqtr4id 2791	. 2 ⊢ (𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
13		0rest 17361	. . 3 ⊢ (∅ ↾t (𝐵 × 𝐵)) = ∅
14		fvprc 6834	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
15	1, 14	eqtrid 2784	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
16	15	oveq1d 7383	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17		fvprc 6834	. . 3 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
18	13, 16, 17	3eqtr4a 2798	. 2 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
19	12, 18	pm2.61i 182	1 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287   × cxp 5630  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  UnifSetcunif 17199   ↾_t crest 17352  UnifStcuss 24209

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-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-rest 17354  df-uss 24212

This theorem is referenced by:  ussid  24216  ressuss  24218