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Theorem ustuqtoplem 23744
Description: Lemma for ustuqtop 23751. (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
ustuqtoplem (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ 𝐴 ∈ 𝑉) β†’ (𝐴 ∈ (π‘β€˜π‘ƒ) ↔ βˆƒπ‘€ ∈ π‘ˆ 𝐴 = (𝑀 β€œ {𝑃})))
Distinct variable groups:   𝑀,𝐴   𝑀,𝑣,𝑃   𝑣,𝑝,𝑀,π‘ˆ   𝑋,𝑝,𝑣
Allowed substitution hints:   𝐴(𝑣,𝑝)   𝑃(𝑝)   𝑁(𝑀,𝑣,𝑝)   𝑉(𝑀,𝑣,𝑝)   𝑋(𝑀)
Proof of Theorem ustuqtoplem
Dummy variable π‘ž is distinct from all other variables.
StepHypRef Expression
1 utopustuq.1 . . . . 5 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
2 simpl 484 . . . . . . . . . 10 ((𝑝 = π‘ž ∧ 𝑣 ∈ π‘ˆ) β†’ 𝑝 = π‘ž)
32sneqd 4641 . . . . . . . . 9 ((𝑝 = π‘ž ∧ 𝑣 ∈ π‘ˆ) β†’ {𝑝} = {π‘ž})
43imaeq2d 6060 . . . . . . . 8 ((𝑝 = π‘ž ∧ 𝑣 ∈ π‘ˆ) β†’ (𝑣 β€œ {𝑝}) = (𝑣 β€œ {π‘ž}))
54mpteq2dva 5249 . . . . . . 7 (𝑝 = π‘ž β†’ (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) = (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {π‘ž})))
65rneqd 5938 . . . . . 6 (𝑝 = π‘ž β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {π‘ž})))
76cbvmptv 5262 . . . . 5 (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝}))) = (π‘ž ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {π‘ž})))
81, 7eqtri 2761 . . . 4 𝑁 = (π‘ž ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {π‘ž})))
9 simpr2 1196 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ (𝑃 ∈ 𝑋 ∧ π‘ž = 𝑃 ∧ 𝑣 ∈ π‘ˆ)) β†’ π‘ž = 𝑃)
109sneqd 4641 . . . . . . . 8 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ (𝑃 ∈ 𝑋 ∧ π‘ž = 𝑃 ∧ 𝑣 ∈ π‘ˆ)) β†’ {π‘ž} = {𝑃})
1110imaeq2d 6060 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ (𝑃 ∈ 𝑋 ∧ π‘ž = 𝑃 ∧ 𝑣 ∈ π‘ˆ)) β†’ (𝑣 β€œ {π‘ž}) = (𝑣 β€œ {𝑃}))
12113anassrs 1361 . . . . . 6 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ π‘ž = 𝑃) ∧ 𝑣 ∈ π‘ˆ) β†’ (𝑣 β€œ {π‘ž}) = (𝑣 β€œ {𝑃}))
1312mpteq2dva 5249 . . . . 5 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ π‘ž = 𝑃) β†’ (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {π‘ž})) = (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
1413rneqd 5938 . . . 4 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ π‘ž = 𝑃) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {π‘ž})) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
15 simpr 486 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ 𝑃 ∈ 𝑋)
16 mptexg 7223 . . . . . 6 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
17 rnexg 7895 . . . . . 6 ((𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
1816, 17syl 17 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
1918adantr 482 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
208, 14, 15, 19fvmptd2 7007 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (π‘β€˜π‘ƒ) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
2120eleq2d 2820 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (𝐴 ∈ (π‘β€˜π‘ƒ) ↔ 𝐴 ∈ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃}))))
22 imaeq1 6055 . . . 4 (𝑣 = 𝑀 β†’ (𝑣 β€œ {𝑃}) = (𝑀 β€œ {𝑃}))
2322cbvmptv 5262 . . 3 (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) = (𝑀 ∈ π‘ˆ ↦ (𝑀 β€œ {𝑃}))
2423elrnmpt 5956 . 2 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ↔ βˆƒπ‘€ ∈ π‘ˆ 𝐴 = (𝑀 β€œ {𝑃})))
2521, 24sylan9bb 511 1 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ 𝐴 ∈ 𝑉) β†’ (𝐴 ∈ (π‘β€˜π‘ƒ) ↔ βˆƒπ‘€ ∈ π‘ˆ 𝐴 = (𝑀 β€œ {𝑃})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475  {csn 4629   ↦ cmpt 5232  ran crn 5678   β€œ cima 5680  β€˜cfv 6544  UnifOncust 23704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552
This theorem is referenced by:  ustuqtop1  23746  ustuqtop2  23747  ustuqtop3  23748  ustuqtop4  23749  ustuqtop5  23750  utopsnneiplem  23752
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