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Theorem brdomaing 34566
Description: Closed form of brdomain 34564. (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 5109 . . 3 (𝑎 = 𝐴 → (𝑎Domain𝑏𝐴Domain𝑏))
2 dmeq 5860 . . . 4 (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
32eqeq2d 2744 . . 3 (𝑎 = 𝐴 → (𝑏 = dom 𝑎𝑏 = dom 𝐴))
41, 3bibi12d 346 . 2 (𝑎 = 𝐴 → ((𝑎Domain𝑏𝑏 = dom 𝑎) ↔ (𝐴Domain𝑏𝑏 = dom 𝐴)))
5 breq2 5110 . . 3 (𝑏 = 𝐵 → (𝐴Domain𝑏𝐴Domain𝐵))
6 eqeq1 2737 . . 3 (𝑏 = 𝐵 → (𝑏 = dom 𝐴𝐵 = dom 𝐴))
75, 6bibi12d 346 . 2 (𝑏 = 𝐵 → ((𝐴Domain𝑏𝑏 = dom 𝐴) ↔ (𝐴Domain𝐵𝐵 = dom 𝐴)))
8 vex 3448 . . 3 𝑎 ∈ V
9 vex 3448 . . 3 𝑏 ∈ V
108, 9brdomain 34564 . 2 (𝑎Domain𝑏𝑏 = dom 𝑎)
114, 7, 10vtocl2g 3530 1 ((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5106  dom cdm 5634  Domaincdomain 34474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-symdif 4203  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-eprel 5538  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922  df-2nd 7923  df-txp 34485  df-image 34495  df-domain 34498
This theorem is referenced by: (None)
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