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Theorem diadm 40508
Description: Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)
Hypotheses
Ref Expression
diafn.b 𝐡 = (Baseβ€˜πΎ)
diafn.l ≀ = (leβ€˜πΎ)
diafn.h 𝐻 = (LHypβ€˜πΎ)
diafn.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diadm ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ dom 𝐼 = {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š})
Distinct variable groups:   π‘₯, ≀   π‘₯,𝐡   π‘₯,𝐾   π‘₯,π‘Š
Allowed substitution hints:   𝐻(π‘₯)   𝐼(π‘₯)   𝑉(π‘₯)

Proof of Theorem diadm
StepHypRef Expression
1 diafn.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 diafn.l . . 3 ≀ = (leβ€˜πΎ)
3 diafn.h . . 3 𝐻 = (LHypβ€˜πΎ)
4 diafn.i . . 3 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
51, 2, 3, 4diafn 40507 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š})
65fndmd 6659 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ dom 𝐼 = {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3429   class class class wbr 5148  dom cdm 5678  β€˜cfv 6548  Basecbs 17179  lecple 17239  LHypclh 39457  DIsoAcdia 40501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-disoa 40502
This theorem is referenced by:  diaeldm  40509  diaglbN  40528  diaintclN  40531  dibfnN  40629  dibglbN  40639
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