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Theorem ustuqtoplem 22821
Description: Lemma for ustuqtop 22828. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
ustuqtoplem (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝐴𝑉) → (𝐴 ∈ (𝑁𝑃) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
Distinct variable groups:   𝑤,𝐴   𝑤,𝑣,𝑃   𝑣,𝑝,𝑤,𝑈   𝑋,𝑝,𝑣
Allowed substitution hints:   𝐴(𝑣,𝑝)   𝑃(𝑝)   𝑁(𝑤,𝑣,𝑝)   𝑉(𝑤,𝑣,𝑝)   𝑋(𝑤)

Proof of Theorem ustuqtoplem
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 utopustuq.1 . . . . 5 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2 simpl 485 . . . . . . . . . 10 ((𝑝 = 𝑞𝑣𝑈) → 𝑝 = 𝑞)
32sneqd 4553 . . . . . . . . 9 ((𝑝 = 𝑞𝑣𝑈) → {𝑝} = {𝑞})
43imaeq2d 5903 . . . . . . . 8 ((𝑝 = 𝑞𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑞}))
54mpteq2dva 5135 . . . . . . 7 (𝑝 = 𝑞 → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
65rneqd 5782 . . . . . 6 (𝑝 = 𝑞 → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
76cbvmptv 5143 . . . . 5 (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))) = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
81, 7eqtri 2843 . . . 4 𝑁 = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
9 simpr2 1191 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → 𝑞 = 𝑃)
109sneqd 4553 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → {𝑞} = {𝑃})
1110imaeq2d 5903 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
12113anassrs 1356 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
1312mpteq2dva 5135 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
1413rneqd 5782 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
15 simpr 487 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
16 mptexg 6958 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
17 rnexg 7590 . . . . . 6 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
1816, 17syl 17 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
1918adantr 483 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
208, 14, 15, 19fvmptd2 6750 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
2120eleq2d 2896 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝐴 ∈ (𝑁𝑃) ↔ 𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
22 imaeq1 5898 . . . 4 (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
2322cbvmptv 5143 . . 3 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑤𝑈 ↦ (𝑤 “ {𝑃}))
2423elrnmpt 5802 . 2 (𝐴𝑉 → (𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
2521, 24sylan9bb 512 1 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝐴𝑉) → (𝐴 ∈ (𝑁𝑃) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wrex 3126  Vcvv 3473  {csn 4541  cmpt 5120  ran crn 5530  cima 5532  cfv 6329  UnifOncust 22781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337
This theorem is referenced by:  ustuqtop1  22823  ustuqtop2  22824  ustuqtop3  22825  ustuqtop4  22826  ustuqtop5  22827  utopsnneiplem  22829
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