Theorem ussval 22868

Description: The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6663 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 6670	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊))
2		fveq2 6670	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3	2	sqxpeqd 5587	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
4	1, 3	oveq12d 7174	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
5		df-uss 22865	. . . 4 ⊢ UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))))
6		ovex 7189	. . . 4 ⊢ ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V
7	4, 5, 6	fvmpt 6768	. . 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 5583	. . . 4 ⊢ (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
11	8, 10	oveq12i 7168	. . 3 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))
12	7, 11	syl6reqr 2875	. 2 ⊢ (𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
13		0rest 16703	. . 3 ⊢ (∅ ↾t (𝐵 × 𝐵)) = ∅
14		fvprc 6663	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
15	8, 14	syl5eq 2868	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
16	15	oveq1d 7171	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17		fvprc 6663	. . 3 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
18	13, 16, 17	3eqtr4a 2882	. 2 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
19	12, 18	pm2.61i 184	1 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ∅c0 4291   × cxp 5553  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  UnifSetcunif 16575   ↾_t crest 16694  UnifStcuss 22862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-rest 16696  df-uss 22865

This theorem is referenced by:  ussid  22869  ressuss  22872