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Theorem ustimasn 24238
Description: Lemma for ustuqtop 24256. (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 6088 . 2 (𝑉 “ {𝑃}) ⊆ ran 𝑉
2 ustssxp 24214 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
323adant3 1132 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → 𝑉 ⊆ (𝑋 × 𝑋))
4 rnss 5949 . . . 4 (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉 ⊆ ran (𝑋 × 𝑋))
5 rnxpid 6192 . . . 4 ran (𝑋 × 𝑋) = 𝑋
64, 5sseqtrdi 4023 . . 3 (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉𝑋)
73, 6syl 17 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ran 𝑉𝑋)
81, 7sstrid 3994 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2107  wss 3950  {csn 4625   × cxp 5682  ran crn 5685  cima 5687  cfv 6560  UnifOncust 24209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ust 24210
This theorem is referenced by:  ustuqtop0  24250  ustuqtop4  24254  utopreg  24262  ucncn  24295
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