MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustimasn Structured version   Visualization version   GIF version

Theorem ustimasn 22254
Description: Lemma for ustuqtop 22272. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
ustimasn ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)

Proof of Theorem ustimasn
StepHypRef Expression
1 imassrn 5636 . 2 (𝑉 “ {𝑃}) ⊆ ran 𝑉
2 ustssxp 22230 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
323adant3 1127 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → 𝑉 ⊆ (𝑋 × 𝑋))
4 rnss 5510 . . . 4 (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉 ⊆ ran (𝑋 × 𝑋))
5 rnxpid 5726 . . . 4 ran (𝑋 × 𝑋) = 𝑋
64, 5syl6sseq 3793 . . 3 (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉𝑋)
73, 6syl 17 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ran 𝑉𝑋)
81, 7syl5ss 3756 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072  wcel 2140  wss 3716  {csn 4322   × cxp 5265  ran crn 5268  cima 5270  cfv 6050  UnifOncust 22225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fv 6058  df-ust 22226
This theorem is referenced by:  ustuqtop0  22266  ustuqtop4  22270  utopreg  22278  ucncn  22311
  Copyright terms: Public domain W3C validator