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Theorem brdomain 33281
 Description: Binary relation form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brdomain.1 𝐴 ∈ V
brdomain.2 𝐵 ∈ V
Assertion
Ref Expression
brdomain (𝐴Domain𝐵𝐵 = dom 𝐴)

Proof of Theorem brdomain
StepHypRef Expression
1 brdomain.1 . . 3 𝐴 ∈ V
2 brdomain.2 . . 3 𝐵 ∈ V
31, 2brimage 33274 . 2 (𝐴Image(1st ↾ (V × V))𝐵𝐵 = ((1st ↾ (V × V)) “ 𝐴))
4 df-domain 33215 . . 3 Domain = Image(1st ↾ (V × V))
54breqi 5069 . 2 (𝐴Domain𝐵𝐴Image(1st ↾ (V × V))𝐵)
6 dfdm5 32903 . . 3 dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
76eqeq2i 2839 . 2 (𝐵 = dom 𝐴𝐵 = ((1st ↾ (V × V)) “ 𝐴))
83, 5, 73bitr4i 304 1 (𝐴Domain𝐵𝐵 = dom 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   = wceq 1530   ∈ wcel 2107  Vcvv 3500   class class class wbr 5063   × cxp 5552  dom cdm 5554   ↾ cres 5556   “ cima 5557  1st c1st 7678  Imagecimage 33188  Domaincdomain 33191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-symdif 4223  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-eprel 5464  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fo 6358  df-fv 6360  df-1st 7680  df-2nd 7681  df-txp 33202  df-image 33212  df-domain 33215 This theorem is referenced by:  brdomaing  33283  dfrecs2  33298  dfrdg4  33299
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