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Theorem ustimasn 23578
Description: Lemma for ustuqtop 23596. (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 6024 . 2 (𝑉 “ {𝑃}) ⊆ ran 𝑉
2 ustssxp 23554 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
323adant3 1132 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → 𝑉 ⊆ (𝑋 × 𝑋))
4 rnss 5894 . . . 4 (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉 ⊆ ran (𝑋 × 𝑋))
5 rnxpid 6125 . . . 4 ran (𝑋 × 𝑋) = 𝑋
64, 5sseqtrdi 3994 . . 3 (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉𝑋)
73, 6syl 17 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ran 𝑉𝑋)
81, 7sstrid 3955 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2106  wss 3910  {csn 4586   × cxp 5631  ran crn 5634  cima 5636  cfv 6496  UnifOncust 23549
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-ust 23550
This theorem is referenced by:  ustuqtop0  23590  ustuqtop4  23594  utopreg  23602  ucncn  23635
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