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Theorem ustuqtop2 23594
Description: Lemma for ustuqtop 23598. (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 23559 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑢𝑈) → (𝑤𝑢) ∈ 𝑈)
62, 3, 4, 5syl3anc 1371 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑤𝑢) ∈ 𝑈)
7 ineq12 4167 . . . . . . . . . . 11 ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
8 inimasn 6108 . . . . . . . . . . . 12 (𝑝 ∈ V → ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
98elv 3451 . . . . . . . . . . 11 ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))
107, 9eqtr4di 2794 . . . . . . . . . 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 6008 . . . . . . . . . 10 (𝑥 = (𝑤𝑢) → (𝑥 “ {𝑝}) = ((𝑤𝑢) “ {𝑝}))
1312rspceeqv 3595 . . . . . . . . 9 (((𝑤𝑢) ∈ 𝑈 ∧ (𝑎𝑏) = ((𝑤𝑢) “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
146, 11, 13syl2anc 584 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
15 vex 3449 . . . . . . . . . . 11 𝑎 ∈ V
1615inex1 5274 . . . . . . . . . 10 (𝑎𝑏) ∈ V
17 utopustuq.1 . . . . . . . . . . 11 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1817ustuqtoplem 23591 . . . . . . . . . 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 478 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
211, 14, 20syl2anc 584 . . . . . . 7 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
2217ustuqtoplem 23591 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
2322elvd 3452 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
2423biimpa 477 . . . . . . . 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 3159 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
2717ustuqtoplem 23591 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2827elvd 3452 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2928biimpa 477 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3029adantr 481 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3126, 30r19.29a 3159 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → (𝑎𝑏) ∈ (𝑁𝑝))
3231ralrimiva 3143 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
3332ralrimiva 3143 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
34 fvex 6855 . . . 4 (𝑁𝑝) ∈ V
35 inficl 9361 . . . 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 4000 . 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 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cin 3909  wss 3910  {csn 4586  cmpt 5188  ran crn 5634  cima 5636  cfv 6496  ficfi 9346  UnifOncust 23551
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-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 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-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  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-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-er 8648  df-en 8884  df-fin 8887  df-fi 9347  df-ust 23552
This theorem is referenced by:  ustuqtop  23598  utopsnneiplem  23599
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