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Theorem brdomaing 32486
Description: Closed form of brdomain 32484. (Contributed by Scott Fenton, 2-May-2014.)
Assertion
Ref Expression
brdomaing ((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))

Proof of Theorem brdomaing
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4812 . . 3 (𝑎 = 𝐴 → (𝑎Domain𝑏𝐴Domain𝑏))
2 dmeq 5492 . . . 4 (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
32eqeq2d 2775 . . 3 (𝑎 = 𝐴 → (𝑏 = dom 𝑎𝑏 = dom 𝐴))
41, 3bibi12d 336 . 2 (𝑎 = 𝐴 → ((𝑎Domain𝑏𝑏 = dom 𝑎) ↔ (𝐴Domain𝑏𝑏 = dom 𝐴)))
5 breq2 4813 . . 3 (𝑏 = 𝐵 → (𝐴Domain𝑏𝐴Domain𝐵))
6 eqeq1 2769 . . 3 (𝑏 = 𝐵 → (𝑏 = dom 𝐴𝐵 = dom 𝐴))
75, 6bibi12d 336 . 2 (𝑏 = 𝐵 → ((𝐴Domain𝑏𝑏 = dom 𝐴) ↔ (𝐴Domain𝐵𝐵 = dom 𝐴)))
8 vex 3353 . . 3 𝑎 ∈ V
9 vex 3353 . . 3 𝑏 ∈ V
108, 9brdomain 32484 . 2 (𝑎Domain𝑏𝑏 = dom 𝑎)
114, 7, 10vtocl2g 3422 1 ((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155   class class class wbr 4809  dom cdm 5277  Domaincdomain 32394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-symdif 4005  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-eprel 5190  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-1st 7366  df-2nd 7367  df-txp 32405  df-image 32415  df-domain 32418
This theorem is referenced by: (None)
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