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Theorem ustuqtoplem 23751
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
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 483 . . . . . . . . . 10 ((𝑝 = π‘ž ∧ 𝑣 ∈ π‘ˆ) β†’ 𝑝 = π‘ž)
32sneqd 4640 . . . . . . . . 9 ((𝑝 = π‘ž ∧ 𝑣 ∈ π‘ˆ) β†’ {𝑝} = {π‘ž})
43imaeq2d 6059 . . . . . . . 8 ((𝑝 = π‘ž ∧ 𝑣 ∈ π‘ˆ) β†’ (𝑣 β€œ {𝑝}) = (𝑣 β€œ {π‘ž}))
54mpteq2dva 5248 . . . . . . 7 (𝑝 = π‘ž β†’ (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) = (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {π‘ž})))
65rneqd 5937 . . . . . 6 (𝑝 = π‘ž β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {π‘ž})))
76cbvmptv 5261 . . . . 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 2760 . . . 4 𝑁 = (π‘ž ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {π‘ž})))
9 simpr2 1195 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ (𝑃 ∈ 𝑋 ∧ π‘ž = 𝑃 ∧ 𝑣 ∈ π‘ˆ)) β†’ π‘ž = 𝑃)
109sneqd 4640 . . . . . . . 8 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ (𝑃 ∈ 𝑋 ∧ π‘ž = 𝑃 ∧ 𝑣 ∈ π‘ˆ)) β†’ {π‘ž} = {𝑃})
1110imaeq2d 6059 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ (𝑃 ∈ 𝑋 ∧ π‘ž = 𝑃 ∧ 𝑣 ∈ π‘ˆ)) β†’ (𝑣 β€œ {π‘ž}) = (𝑣 β€œ {𝑃}))
12113anassrs 1360 . . . . . 6 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ π‘ž = 𝑃) ∧ 𝑣 ∈ π‘ˆ) β†’ (𝑣 β€œ {π‘ž}) = (𝑣 β€œ {𝑃}))
1312mpteq2dva 5248 . . . . 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 5937 . . . 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 485 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ 𝑃 ∈ 𝑋)
16 mptexg 7225 . . . . . 6 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
17 rnexg 7897 . . . . . 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 481 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
208, 14, 15, 19fvmptd2 7006 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (π‘β€˜π‘ƒ) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
2120eleq2d 2819 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (𝐴 ∈ (π‘β€˜π‘ƒ) ↔ 𝐴 ∈ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃}))))
22 imaeq1 6054 . . . 4 (𝑣 = 𝑀 β†’ (𝑣 β€œ {𝑃}) = (𝑀 β€œ {𝑃}))
2322cbvmptv 5261 . . 3 (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) = (𝑀 ∈ π‘ˆ ↦ (𝑀 β€œ {𝑃}))
2423elrnmpt 5955 . 2 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ↔ βˆƒπ‘€ ∈ π‘ˆ 𝐴 = (𝑀 β€œ {𝑃})))
2521, 24sylan9bb 510 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474  {csn 4628   ↦ cmpt 5231  ran crn 5677   β€œ cima 5679  β€˜cfv 6543  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-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  ustuqtop1  23753  ustuqtop2  23754  ustuqtop3  23755  ustuqtop4  23756  ustuqtop5  23757  utopsnneiplem  23759
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