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Theorem ustimasn 22252
Description: Lemma for ustuqtop 22270. (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 5618 . 2 (𝑉 “ {𝑃}) ⊆ ran 𝑉
2 ustssxp 22228 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
323adant3 1126 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → 𝑉 ⊆ (𝑋 × 𝑋))
4 rnss 5492 . . . 4 (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉 ⊆ ran (𝑋 × 𝑋))
5 rnxpid 5708 . . . 4 ran (𝑋 × 𝑋) = 𝑋
64, 5syl6sseq 3800 . . 3 (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉𝑋)
73, 6syl 17 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ran 𝑉𝑋)
81, 7syl5ss 3763 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071  wcel 2145  wss 3723  {csn 4316   × cxp 5247  ran crn 5250  cima 5252  cfv 6031  UnifOncust 22223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ust 22224
This theorem is referenced by:  ustuqtop0  22264  ustuqtop4  22268  utopreg  22276  ucncn  22309
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