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Theorem dibdmN 40539
Description: Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibfn.b 𝐡 = (Baseβ€˜πΎ)
dibfn.l ≀ = (leβ€˜πΎ)
dibfn.h 𝐻 = (LHypβ€˜πΎ)
dibfn.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibdmN ((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 dibdmN
StepHypRef Expression
1 dibfn.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 dibfn.l . . 3 ≀ = (leβ€˜πΎ)
3 dibfn.h . . 3 𝐻 = (LHypβ€˜πΎ)
4 dibfn.i . . 3 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
51, 2, 3, 4dibfnN 40538 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š})
65fndmd 6647 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 1533   ∈ wcel 2098  {crab 3426   class class class wbr 5141  dom cdm 5669  β€˜cfv 6536  Basecbs 17151  lecple 17211  LHypclh 39366  DIsoBcdib 40520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-disoa 40411  df-dib 40521
This theorem is referenced by:  dibglbN  40548  dibintclN  40549  dihglblem3N  40677
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