Theorem ussval 24571

Description: The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6875 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 5679	. . . 4 ⊢ (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
4	1, 3	oveq12i 7430	. . 3 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))
5		fveq2 6883	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊))
6		fveq2 6883	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7	6	sqxpeqd 5683	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
8	5, 7	oveq12d 7436	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
9		df-uss 24568	. . . 4 ⊢ UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))))
10		ovex 7451	. . . 4 ⊢ ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V
11	8, 9, 10	fvmpt 6991	. . 3 ⊢ (𝑊 ∈ V → (UnifSt‘𝑊) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
12	4, 11	eqtr4id 2815	. 2 ⊢ (𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
13		0rest 17593	. . 3 ⊢ (∅ ↾t (𝐵 × 𝐵)) = ∅
14		fvprc 6875	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
15	1, 14	eqtrid 2808	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
16	15	oveq1d 7433	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17		fvprc 6875	. . 3 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
18	13, 16, 17	3eqtr4a 2822	. 2 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
19	12, 18	pm2.61i 184	1 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279   × cxp 5649  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  UnifSetcunif 17431   ↾_t crest 17584  UnifStcuss 24565

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-rest 17586  df-uss 24568

This theorem is used by:  ussid  24572  ressuss  24574