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Theorem ustuqtop2 24196
Description: Lemma for ustuqtop 24200. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
ustuqtop2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣   𝑁,𝑝
Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop2
Dummy variables 𝑤 𝑎 𝑏 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp-6l 785 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
2 simp-7l 787 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3 simp-4r 782 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑤𝑈)
4 simplr 767 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑢𝑈)
5 ustincl 24161 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑢𝑈) → (𝑤𝑢) ∈ 𝑈)
62, 3, 4, 5syl3anc 1368 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑤𝑢) ∈ 𝑈)
7 ineq12 4205 . . . . . . . . . . 11 ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
8 inimasn 6162 . . . . . . . . . . . 12 (𝑝 ∈ V → ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
98elv 3467 . . . . . . . . . . 11 ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))
107, 9eqtr4di 2783 . . . . . . . . . 10 ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤𝑢) “ {𝑝}))
1110ad4ant24 752 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤𝑢) “ {𝑝}))
12 imaeq1 6059 . . . . . . . . . 10 (𝑥 = (𝑤𝑢) → (𝑥 “ {𝑝}) = ((𝑤𝑢) “ {𝑝}))
1312rspceeqv 3628 . . . . . . . . 9 (((𝑤𝑢) ∈ 𝑈 ∧ (𝑎𝑏) = ((𝑤𝑢) “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
146, 11, 13syl2anc 582 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
15 vex 3465 . . . . . . . . . . 11 𝑎 ∈ V
1615inex1 5318 . . . . . . . . . 10 (𝑎𝑏) ∈ V
17 utopustuq.1 . . . . . . . . . . 11 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1817ustuqtoplem 24193 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑎𝑏) ∈ V) → ((𝑎𝑏) ∈ (𝑁𝑝) ↔ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})))
1916, 18mpan2 689 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑎𝑏) ∈ (𝑁𝑝) ↔ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})))
2019biimpar 476 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
211, 14, 20syl2anc 582 . . . . . . 7 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
2217ustuqtoplem 24193 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
2322elvd 3468 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
2423biimpa 475 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
2524ad5ant13 755 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
2621, 25r19.29a 3151 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
2717ustuqtoplem 24193 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2827elvd 3468 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2928biimpa 475 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3029adantr 479 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3126, 30r19.29a 3151 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → (𝑎𝑏) ∈ (𝑁𝑝))
3231ralrimiva 3135 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
3332ralrimiva 3135 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
34 fvex 6909 . . . 4 (𝑁𝑝) ∈ V
35 inficl 9455 . . . 4 ((𝑁𝑝) ∈ V → (∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝) ↔ (fi‘(𝑁𝑝)) = (𝑁𝑝)))
3634, 35ax-mp 5 . . 3 (∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝) ↔ (fi‘(𝑁𝑝)) = (𝑁𝑝))
3733, 36sylib 217 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) = (𝑁𝑝))
38 eqimss 4035 . 2 ((fi‘(𝑁𝑝)) = (𝑁𝑝) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
3937, 38syl 17 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  cin 3943  wss 3944  {csn 4630  cmpt 5232  ran crn 5679  cima 5681  cfv 6549  ficfi 9440  UnifOncust 24153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-2o 8488  df-en 8965  df-fin 8968  df-fi 9441  df-ust 24154
This theorem is referenced by:  ustuqtop  24200  utopsnneiplem  24201
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