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Theorem dibdmN 37824
Description: Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibfn.b 𝐵 = (Base‘𝐾)
dibfn.l = (le‘𝐾)
dibfn.h 𝐻 = (LHyp‘𝐾)
dibfn.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibdmN ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
Distinct variable groups:   𝑥,   𝑥,𝐵   𝑥,𝐾   𝑥,𝑊
Allowed substitution hints:   𝐻(𝑥)   𝐼(𝑥)   𝑉(𝑥)

Proof of Theorem dibdmN
StepHypRef Expression
1 dibfn.b . . 3 𝐵 = (Base‘𝐾)
2 dibfn.l . . 3 = (le‘𝐾)
3 dibfn.h . . 3 𝐻 = (LHyp‘𝐾)
4 dibfn.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
51, 2, 3, 4dibfnN 37823 . 2 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
6 fndm 6325 . 2 (𝐼 Fn {𝑥𝐵𝑥 𝑊} → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
75, 6syl 17 1 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  {crab 3109   class class class wbr 4962  dom cdm 5443   Fn wfn 6220  cfv 6225  Basecbs 16312  lecple 16401  LHypclh 36651  DIsoBcdib 37805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-disoa 37696  df-dib 37806
This theorem is referenced by:  dibglbN  37833  dibintclN  37834  dihglblem3N  37962
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