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Theorem dilfsetN 38618
Description: The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
dilset.a 𝐴 = (Atomsβ€˜πΎ)
dilset.s 𝑆 = (PSubSpβ€˜πΎ)
dilset.w π‘Š = (WAtomsβ€˜πΎ)
dilset.m 𝑀 = (PAutβ€˜πΎ)
dilset.l 𝐿 = (Dilβ€˜πΎ)
Assertion
Ref Expression
dilfsetN (𝐾 ∈ 𝐡 β†’ 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
Distinct variable groups:   𝐴,𝑑   𝑓,𝑑,π‘₯,𝐾   𝑓,𝑀   π‘₯,𝑆
Allowed substitution hints:   𝐴(π‘₯,𝑓)   𝐡(π‘₯,𝑓,𝑑)   𝑆(𝑓,𝑑)   𝐿(π‘₯,𝑓,𝑑)   𝑀(π‘₯,𝑑)   π‘Š(π‘₯,𝑓,𝑑)

Proof of Theorem dilfsetN
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3464 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 dilset.l . . 3 𝐿 = (Dilβ€˜πΎ)
3 fveq2 6843 . . . . . 6 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 dilset.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2795 . . . . 5 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
6 fveq2 6843 . . . . . . 7 (π‘˜ = 𝐾 β†’ (PAutβ€˜π‘˜) = (PAutβ€˜πΎ))
7 dilset.m . . . . . . 7 𝑀 = (PAutβ€˜πΎ)
86, 7eqtr4di 2795 . . . . . 6 (π‘˜ = 𝐾 β†’ (PAutβ€˜π‘˜) = 𝑀)
9 fveq2 6843 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (PSubSpβ€˜π‘˜) = (PSubSpβ€˜πΎ))
10 dilset.s . . . . . . . 8 𝑆 = (PSubSpβ€˜πΎ)
119, 10eqtr4di 2795 . . . . . . 7 (π‘˜ = 𝐾 β†’ (PSubSpβ€˜π‘˜) = 𝑆)
12 fveq2 6843 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (WAtomsβ€˜π‘˜) = (WAtomsβ€˜πΎ))
13 dilset.w . . . . . . . . . . 11 π‘Š = (WAtomsβ€˜πΎ)
1412, 13eqtr4di 2795 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (WAtomsβ€˜π‘˜) = π‘Š)
1514fveq1d 6845 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ ((WAtomsβ€˜π‘˜)β€˜π‘‘) = (π‘Šβ€˜π‘‘))
1615sseq2d 3977 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (π‘₯ βŠ† ((WAtomsβ€˜π‘˜)β€˜π‘‘) ↔ π‘₯ βŠ† (π‘Šβ€˜π‘‘)))
1716imbi1d 342 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((π‘₯ βŠ† ((WAtomsβ€˜π‘˜)β€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)))
1811, 17raleqbidv 3320 . . . . . 6 (π‘˜ = 𝐾 β†’ (βˆ€π‘₯ ∈ (PSubSpβ€˜π‘˜)(π‘₯ βŠ† ((WAtomsβ€˜π‘˜)β€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)))
198, 18rabeqbidv 3425 . . . . 5 (π‘˜ = 𝐾 β†’ {𝑓 ∈ (PAutβ€˜π‘˜) ∣ βˆ€π‘₯ ∈ (PSubSpβ€˜π‘˜)(π‘₯ βŠ† ((WAtomsβ€˜π‘˜)β€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)} = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)})
205, 19mpteq12dv 5197 . . . 4 (π‘˜ = 𝐾 β†’ (𝑑 ∈ (Atomsβ€˜π‘˜) ↦ {𝑓 ∈ (PAutβ€˜π‘˜) ∣ βˆ€π‘₯ ∈ (PSubSpβ€˜π‘˜)(π‘₯ βŠ† ((WAtomsβ€˜π‘˜)β€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
21 df-dilN 38572 . . . 4 Dil = (π‘˜ ∈ V ↦ (𝑑 ∈ (Atomsβ€˜π‘˜) ↦ {𝑓 ∈ (PAutβ€˜π‘˜) ∣ βˆ€π‘₯ ∈ (PSubSpβ€˜π‘˜)(π‘₯ βŠ† ((WAtomsβ€˜π‘˜)β€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
2220, 21, 4mptfvmpt 7179 . . 3 (𝐾 ∈ V β†’ (Dilβ€˜πΎ) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
232, 22eqtrid 2789 . 2 (𝐾 ∈ V β†’ 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
241, 23syl 17 1 (𝐾 ∈ 𝐡 β†’ 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408  Vcvv 3446   βŠ† wss 3911   ↦ cmpt 5189  β€˜cfv 6497  Atomscatm 37728  PSubSpcpsubsp 37962  WAtomscwpointsN 38452  PAutcpautN 38453  DilcdilN 38568
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-dilN 38572
This theorem is referenced by:  dilsetN  38619
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