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Theorem isdilN 38620
Description: The predicate "is a dilation". (Contributed by NM, 4-Feb-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
isdilN ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (πΏβ€˜π·) ↔ (𝐹 ∈ 𝑀 ∧ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (πΉβ€˜π‘₯) = π‘₯))))
Distinct variable groups:   π‘₯,𝐾   π‘₯,𝑆   π‘₯,𝐷   π‘₯,𝐹
Allowed substitution hints:   𝐴(π‘₯)   𝐡(π‘₯)   𝐿(π‘₯)   𝑀(π‘₯)   π‘Š(π‘₯)

Proof of Theorem isdilN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dilset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 dilset.s . . . 4 𝑆 = (PSubSpβ€˜πΎ)
3 dilset.w . . . 4 π‘Š = (WAtomsβ€˜πΎ)
4 dilset.m . . . 4 𝑀 = (PAutβ€˜πΎ)
5 dilset.l . . . 4 𝐿 = (Dilβ€˜πΎ)
61, 2, 3, 4, 5dilsetN 38619 . . 3 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (πΏβ€˜π·) = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
76eleq2d 2824 . 2 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (πΏβ€˜π·) ↔ 𝐹 ∈ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
8 fveq1 6842 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
98eqeq1d 2739 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) = π‘₯ ↔ (πΉβ€˜π‘₯) = π‘₯))
109imbi2d 341 . . . 4 (𝑓 = 𝐹 β†’ ((π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (πΉβ€˜π‘₯) = π‘₯)))
1110ralbidv 3175 . . 3 (𝑓 = 𝐹 β†’ (βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (πΉβ€˜π‘₯) = π‘₯)))
1211elrab 3646 . 2 (𝐹 ∈ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)} ↔ (𝐹 ∈ 𝑀 ∧ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (πΉβ€˜π‘₯) = π‘₯)))
137, 12bitrdi 287 1 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (πΏβ€˜π·) ↔ (𝐹 ∈ 𝑀 ∧ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (πΉβ€˜π‘₯) = π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408   βŠ† wss 3911  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: (None)
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