Theorem ussval 23695

Description: The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6871 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 5698	. . . 4 ⊢ (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
4	1, 3	oveq12i 7406	. . 3 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))
5		fveq2 6879	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊))
6		fveq2 6879	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7	6	sqxpeqd 5702	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
8	5, 7	oveq12d 7412	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
9		df-uss 23692	. . . 4 ⊢ UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))))
10		ovex 7427	. . . 4 ⊢ ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V
11	8, 9, 10	fvmpt 6985	. . 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 17359	. . 3 ⊢ (∅ ↾t (𝐵 × 𝐵)) = ∅
14		fvprc 6871	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
15	1, 14	eqtrid 2784	. . . 4 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
16	15	oveq1d 7409	. . 3 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17		fvprc 6871	. . 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 1541   ∈ wcel 2106  Vcvv 3474  ∅c0 4319   × cxp 5668  ‘cfv 6533  (class class class)co 7394  Basecbs 17128  UnifSetcunif 17191   ↾_t crest 17350  UnifStcuss 23689

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-rest 17352  df-uss 23692

This theorem is referenced by:  ussid  23696  ressuss  23698