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Theorem isdilN 34242
Description: The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
dilset.a 𝐴 = (Atoms‘𝐾)
dilset.s 𝑆 = (PSubSp‘𝐾)
dilset.w 𝑊 = (WAtoms‘𝐾)
dilset.m 𝑀 = (PAut‘𝐾)
dilset.l 𝐿 = (Dil‘𝐾)
Assertion
Ref Expression
isdilN ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝐿𝐷) ↔ (𝐹𝑀 ∧ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥))))
Distinct variable groups:   𝑥,𝐾   𝑥,𝑆   𝑥,𝐷   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐿(𝑥)   𝑀(𝑥)   𝑊(𝑥)

Proof of Theorem isdilN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dilset.a . . . 4 𝐴 = (Atoms‘𝐾)
2 dilset.s . . . 4 𝑆 = (PSubSp‘𝐾)
3 dilset.w . . . 4 𝑊 = (WAtoms‘𝐾)
4 dilset.m . . . 4 𝑀 = (PAut‘𝐾)
5 dilset.l . . . 4 𝐿 = (Dil‘𝐾)
61, 2, 3, 4, 5dilsetN 34241 . . 3 ((𝐾𝐵𝐷𝐴) → (𝐿𝐷) = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)})
76eleq2d 2672 . 2 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝐿𝐷) ↔ 𝐹 ∈ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)}))
8 fveq1 6086 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2611 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑥) = 𝑥 ↔ (𝐹𝑥) = 𝑥))
109imbi2d 328 . . . 4 (𝑓 = 𝐹 → ((𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥)))
1110ralbidv 2968 . . 3 (𝑓 = 𝐹 → (∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥)))
1211elrab 3330 . 2 (𝐹 ∈ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)} ↔ (𝐹𝑀 ∧ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥)))
137, 12syl6bb 274 1 ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝐿𝐷) ↔ (𝐹𝑀 ∧ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  {crab 2899  wss 3539  cfv 5789  Atomscatm 33351  PSubSpcpsubsp 33583  WAtomscwpointsN 34073  PAutcpautN 34074  DilcdilN 34189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-dilN 34193
This theorem is referenced by: (None)
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