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Theorem ustuqtop3 21857
 Description: Lemma for ustuqtop 21860, similar to elnei 20725. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
ustuqtop3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣,𝑎   𝑁,𝑎,𝑝   𝑣,𝑎,𝑈   𝑋,𝑎
Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fnresi 5922 . . . . . . 7 ( I ↾ 𝑋) Fn 𝑋
2 fnsnfv 6168 . . . . . . 7 ((( I ↾ 𝑋) Fn 𝑋𝑝𝑋) → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
31, 2mpan 702 . . . . . 6 (𝑝𝑋 → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
43ad4antlr 765 . . . . 5 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
5 simp-4l 802 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
6 simplr 788 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤𝑈)
7 ustdiag 21822 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → ( I ↾ 𝑋) ⊆ 𝑤)
85, 6, 7syl2anc 691 . . . . . 6 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ( I ↾ 𝑋) ⊆ 𝑤)
9 imass1 5419 . . . . . 6 (( I ↾ 𝑋) ⊆ 𝑤 → (( I ↾ 𝑋) “ {𝑝}) ⊆ (𝑤 “ {𝑝}))
108, 9syl 17 . . . . 5 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (( I ↾ 𝑋) “ {𝑝}) ⊆ (𝑤 “ {𝑝}))
114, 10eqsstrd 3602 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {(( I ↾ 𝑋)‘𝑝)} ⊆ (𝑤 “ {𝑝}))
12 fvex 6113 . . . . 5 (( I ↾ 𝑋)‘𝑝) ∈ V
1312snss 4259 . . . 4 ((( I ↾ 𝑋)‘𝑝) ∈ (𝑤 “ {𝑝}) ↔ {(( I ↾ 𝑋)‘𝑝)} ⊆ (𝑤 “ {𝑝}))
1411, 13sylibr 223 . . 3 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (( I ↾ 𝑋)‘𝑝) ∈ (𝑤 “ {𝑝}))
15 fvresi 6344 . . . . 5 (𝑝𝑋 → (( I ↾ 𝑋)‘𝑝) = 𝑝)
1615eqcomd 2616 . . . 4 (𝑝𝑋𝑝 = (( I ↾ 𝑋)‘𝑝))
1716ad4antlr 765 . . 3 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝 = (( I ↾ 𝑋)‘𝑝))
18 simpr 476 . . 3 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑎 = (𝑤 “ {𝑝}))
1914, 17, 183eltr4d 2703 . 2 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝𝑎)
20 vex 3176 . . . 4 𝑎 ∈ V
21 utopustuq.1 . . . . 5 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2221ustuqtoplem 21853 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2320, 22mpan2 703 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2423biimpa 500 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
2519, 24r19.29a 3060 1 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125   ↦ cmpt 4643   I cid 4948  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ‘cfv 5804  UnifOncust 21813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ust 21814 This theorem is referenced by:  ustuqtop  21860  utopsnneiplem  21861
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