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Theorem ustuqtop3 22417
Description: Lemma for ustuqtop 22420, similar to elnei 21286. (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 6241 . . . . . . 7 ( I ↾ 𝑋) Fn 𝑋
2 fnsnfv 6505 . . . . . . 7 ((( I ↾ 𝑋) Fn 𝑋𝑝𝑋) → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
31, 2mpan 681 . . . . . 6 (𝑝𝑋 → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
43ad4antlr 726 . . . . 5 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {(( I ↾ 𝑋)‘𝑝)} = (( I ↾ 𝑋) “ {𝑝}))
5 simp-4l 801 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
6 simplr 785 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤𝑈)
7 ustdiag 22382 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → ( I ↾ 𝑋) ⊆ 𝑤)
85, 6, 7syl2anc 579 . . . . . 6 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ( I ↾ 𝑋) ⊆ 𝑤)
9 imass1 5741 . . . . . 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 3864 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {(( I ↾ 𝑋)‘𝑝)} ⊆ (𝑤 “ {𝑝}))
12 fvex 6446 . . . . 5 (( I ↾ 𝑋)‘𝑝) ∈ V
1312snss 4535 . . . 4 ((( I ↾ 𝑋)‘𝑝) ∈ (𝑤 “ {𝑝}) ↔ {(( I ↾ 𝑋)‘𝑝)} ⊆ (𝑤 “ {𝑝}))
1411, 13sylibr 226 . . 3 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (( I ↾ 𝑋)‘𝑝) ∈ (𝑤 “ {𝑝}))
15 fvresi 6691 . . . . 5 (𝑝𝑋 → (( I ↾ 𝑋)‘𝑝) = 𝑝)
1615eqcomd 2831 . . . 4 (𝑝𝑋𝑝 = (( I ↾ 𝑋)‘𝑝))
1716ad4antlr 726 . . 3 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝 = (( I ↾ 𝑋)‘𝑝))
18 simpr 479 . . 3 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑎 = (𝑤 “ {𝑝}))
1914, 17, 183eltr4d 2921 . 2 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝𝑎)
20 utopustuq.1 . . . . 5 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2120ustuqtoplem 22413 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2221elvd 3419 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2322biimpa 470 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
2419, 23r19.29a 3288 1 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  wrex 3118  Vcvv 3414  wss 3798  {csn 4397  cmpt 4952   I cid 5249  ran crn 5343  cres 5344  cima 5345   Fn wfn 6118  cfv 6123  UnifOncust 22373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ust 22374
This theorem is referenced by:  ustuqtop  22420  utopsnneiplem  22421
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