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Theorem brdomaing 33883
Description: Closed form of brdomain 33881. (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 5034 . . 3 (𝑎 = 𝐴 → (𝑎Domain𝑏𝐴Domain𝑏))
2 dmeq 5747 . . . 4 (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
32eqeq2d 2750 . . 3 (𝑎 = 𝐴 → (𝑏 = dom 𝑎𝑏 = dom 𝐴))
41, 3bibi12d 349 . 2 (𝑎 = 𝐴 → ((𝑎Domain𝑏𝑏 = dom 𝑎) ↔ (𝐴Domain𝑏𝑏 = dom 𝐴)))
5 breq2 5035 . . 3 (𝑏 = 𝐵 → (𝐴Domain𝑏𝐴Domain𝐵))
6 eqeq1 2743 . . 3 (𝑏 = 𝐵 → (𝑏 = dom 𝐴𝐵 = dom 𝐴))
75, 6bibi12d 349 . 2 (𝑏 = 𝐵 → ((𝐴Domain𝑏𝑏 = dom 𝐴) ↔ (𝐴Domain𝐵𝐵 = dom 𝐴)))
8 vex 3403 . . 3 𝑎 ∈ V
9 vex 3403 . . 3 𝑏 ∈ V
108, 9brdomain 33881 . 2 (𝑎Domain𝑏𝑏 = dom 𝑎)
114, 7, 10vtocl2g 3476 1 ((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114   class class class wbr 5031  dom cdm 5526  Domaincdomain 33791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-symdif 4134  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-eprel 5435  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-fo 6346  df-fv 6348  df-1st 7717  df-2nd 7718  df-txp 33802  df-image 33812  df-domain 33815
This theorem is referenced by: (None)
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