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Theorem brdomaing 32379
Description: Closed form of brdomain 32377. (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 4790 . . 3 (𝑎 = 𝐴 → (𝑎Domain𝑏𝐴Domain𝑏))
2 dmeq 5461 . . . 4 (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
32eqeq2d 2781 . . 3 (𝑎 = 𝐴 → (𝑏 = dom 𝑎𝑏 = dom 𝐴))
41, 3bibi12d 334 . 2 (𝑎 = 𝐴 → ((𝑎Domain𝑏𝑏 = dom 𝑎) ↔ (𝐴Domain𝑏𝑏 = dom 𝐴)))
5 breq2 4791 . . 3 (𝑏 = 𝐵 → (𝐴Domain𝑏𝐴Domain𝐵))
6 eqeq1 2775 . . 3 (𝑏 = 𝐵 → (𝑏 = dom 𝐴𝐵 = dom 𝐴))
75, 6bibi12d 334 . 2 (𝑏 = 𝐵 → ((𝐴Domain𝑏𝑏 = dom 𝐴) ↔ (𝐴Domain𝐵𝐵 = dom 𝐴)))
8 vex 3354 . . 3 𝑎 ∈ V
9 vex 3354 . . 3 𝑏 ∈ V
108, 9brdomain 32377 . 2 (𝑎Domain𝑏𝑏 = dom 𝑎)
114, 7, 10vtocl2g 3421 1 ((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145   class class class wbr 4787  dom cdm 5250  Domaincdomain 32287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-symdif 3994  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-eprel 5163  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fo 6036  df-fv 6038  df-1st 7319  df-2nd 7320  df-txp 32298  df-image 32308  df-domain 32311
This theorem is referenced by: (None)
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