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Theorem ustuqtop2 23754
Description: Lemma for ustuqtop 23758. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
Assertion
Ref Expression
ustuqtop2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (fiβ€˜(π‘β€˜π‘)) βŠ† (π‘β€˜π‘))
Distinct variable groups:   𝑣,𝑝,π‘ˆ   𝑋,𝑝,𝑣   𝑁,𝑝
Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop2
Dummy variables 𝑀 π‘Ž 𝑏 𝑒 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp-6l 785 . . . . . . . 8 ((((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) ∧ 𝑒 ∈ π‘ˆ) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ (π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋))
2 simp-7l 787 . . . . . . . . . 10 ((((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) ∧ 𝑒 ∈ π‘ˆ) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
3 simp-4r 782 . . . . . . . . . 10 ((((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) ∧ 𝑒 ∈ π‘ˆ) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ 𝑀 ∈ π‘ˆ)
4 simplr 767 . . . . . . . . . 10 ((((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) ∧ 𝑒 ∈ π‘ˆ) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ 𝑒 ∈ π‘ˆ)
5 ustincl 23719 . . . . . . . . . 10 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑀 ∈ π‘ˆ ∧ 𝑒 ∈ π‘ˆ) β†’ (𝑀 ∩ 𝑒) ∈ π‘ˆ)
62, 3, 4, 5syl3anc 1371 . . . . . . . . 9 ((((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) ∧ 𝑒 ∈ π‘ˆ) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ (𝑀 ∩ 𝑒) ∈ π‘ˆ)
7 ineq12 4207 . . . . . . . . . . 11 ((π‘Ž = (𝑀 β€œ {𝑝}) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ (π‘Ž ∩ 𝑏) = ((𝑀 β€œ {𝑝}) ∩ (𝑒 β€œ {𝑝})))
8 inimasn 6155 . . . . . . . . . . . 12 (𝑝 ∈ V β†’ ((𝑀 ∩ 𝑒) β€œ {𝑝}) = ((𝑀 β€œ {𝑝}) ∩ (𝑒 β€œ {𝑝})))
98elv 3480 . . . . . . . . . . 11 ((𝑀 ∩ 𝑒) β€œ {𝑝}) = ((𝑀 β€œ {𝑝}) ∩ (𝑒 β€œ {𝑝}))
107, 9eqtr4di 2790 . . . . . . . . . 10 ((π‘Ž = (𝑀 β€œ {𝑝}) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ (π‘Ž ∩ 𝑏) = ((𝑀 ∩ 𝑒) β€œ {𝑝}))
1110ad4ant24 752 . . . . . . . . 9 ((((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) ∧ 𝑒 ∈ π‘ˆ) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ (π‘Ž ∩ 𝑏) = ((𝑀 ∩ 𝑒) β€œ {𝑝}))
12 imaeq1 6054 . . . . . . . . . 10 (π‘₯ = (𝑀 ∩ 𝑒) β†’ (π‘₯ β€œ {𝑝}) = ((𝑀 ∩ 𝑒) β€œ {𝑝}))
1312rspceeqv 3633 . . . . . . . . 9 (((𝑀 ∩ 𝑒) ∈ π‘ˆ ∧ (π‘Ž ∩ 𝑏) = ((𝑀 ∩ 𝑒) β€œ {𝑝})) β†’ βˆƒπ‘₯ ∈ π‘ˆ (π‘Ž ∩ 𝑏) = (π‘₯ β€œ {𝑝}))
146, 11, 13syl2anc 584 . . . . . . . 8 ((((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) ∧ 𝑒 ∈ π‘ˆ) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ βˆƒπ‘₯ ∈ π‘ˆ (π‘Ž ∩ 𝑏) = (π‘₯ β€œ {𝑝}))
15 vex 3478 . . . . . . . . . . 11 π‘Ž ∈ V
1615inex1 5317 . . . . . . . . . 10 (π‘Ž ∩ 𝑏) ∈ V
17 utopustuq.1 . . . . . . . . . . 11 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
1817ustuqtoplem 23751 . . . . . . . . . 10 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ (π‘Ž ∩ 𝑏) ∈ V) β†’ ((π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘) ↔ βˆƒπ‘₯ ∈ π‘ˆ (π‘Ž ∩ 𝑏) = (π‘₯ β€œ {𝑝})))
1916, 18mpan2 689 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ ((π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘) ↔ βˆƒπ‘₯ ∈ π‘ˆ (π‘Ž ∩ 𝑏) = (π‘₯ β€œ {𝑝})))
2019biimpar 478 . . . . . . . 8 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ βˆƒπ‘₯ ∈ π‘ˆ (π‘Ž ∩ 𝑏) = (π‘₯ β€œ {𝑝})) β†’ (π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘))
211, 14, 20syl2anc 584 . . . . . . 7 ((((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) ∧ 𝑒 ∈ π‘ˆ) ∧ 𝑏 = (𝑒 β€œ {𝑝})) β†’ (π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘))
2217ustuqtoplem 23751 . . . . . . . . . 10 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ V) β†’ (𝑏 ∈ (π‘β€˜π‘) ↔ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝})))
2322elvd 3481 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (𝑏 ∈ (π‘β€˜π‘) ↔ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝})))
2423biimpa 477 . . . . . . . 8 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝}))
2524ad5ant13 755 . . . . . . 7 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝}))
2621, 25r19.29a 3162 . . . . . 6 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ (π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘))
2717ustuqtoplem 23751 . . . . . . . . 9 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ V) β†’ (π‘Ž ∈ (π‘β€˜π‘) ↔ βˆƒπ‘€ ∈ π‘ˆ π‘Ž = (𝑀 β€œ {𝑝})))
2827elvd 3481 . . . . . . . 8 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (π‘Ž ∈ (π‘β€˜π‘) ↔ βˆƒπ‘€ ∈ π‘ˆ π‘Ž = (𝑀 β€œ {𝑝})))
2928biimpa 477 . . . . . . 7 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘€ ∈ π‘ˆ π‘Ž = (𝑀 β€œ {𝑝}))
3029adantr 481 . . . . . 6 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘€ ∈ π‘ˆ π‘Ž = (𝑀 β€œ {𝑝}))
3126, 30r19.29a 3162 . . . . 5 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑏 ∈ (π‘β€˜π‘)) β†’ (π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘))
3231ralrimiva 3146 . . . 4 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆ€π‘ ∈ (π‘β€˜π‘)(π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘))
3332ralrimiva 3146 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ βˆ€π‘Ž ∈ (π‘β€˜π‘)βˆ€π‘ ∈ (π‘β€˜π‘)(π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘))
34 fvex 6904 . . . 4 (π‘β€˜π‘) ∈ V
35 inficl 9422 . . . 4 ((π‘β€˜π‘) ∈ V β†’ (βˆ€π‘Ž ∈ (π‘β€˜π‘)βˆ€π‘ ∈ (π‘β€˜π‘)(π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘) ↔ (fiβ€˜(π‘β€˜π‘)) = (π‘β€˜π‘)))
3634, 35ax-mp 5 . . 3 (βˆ€π‘Ž ∈ (π‘β€˜π‘)βˆ€π‘ ∈ (π‘β€˜π‘)(π‘Ž ∩ 𝑏) ∈ (π‘β€˜π‘) ↔ (fiβ€˜(π‘β€˜π‘)) = (π‘β€˜π‘))
3733, 36sylib 217 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (fiβ€˜(π‘β€˜π‘)) = (π‘β€˜π‘))
38 eqimss 4040 . 2 ((fiβ€˜(π‘β€˜π‘)) = (π‘β€˜π‘) β†’ (fiβ€˜(π‘β€˜π‘)) βŠ† (π‘β€˜π‘))
3937, 38syl 17 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (fiβ€˜(π‘β€˜π‘)) βŠ† (π‘β€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  {csn 4628   ↦ cmpt 5231  ran crn 5677   β€œ cima 5679  β€˜cfv 6543  ficfi 9407  UnifOncust 23711
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408  df-ust 23712
This theorem is referenced by:  ustuqtop  23758  utopsnneiplem  23759
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