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Theorem brdomain 33880
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 33873 . 2 (𝐴Image(1st ↾ (V × V))𝐵𝐵 = ((1st ↾ (V × V)) “ 𝐴))
4 df-domain 33814 . . 3 Domain = Image(1st ↾ (V × V))
54breqi 5036 . 2 (𝐴Domain𝐵𝐴Image(1st ↾ (V × V))𝐵)
6 dfdm5 33323 . . 3 dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
76eqeq2i 2751 . 2 (𝐵 = dom 𝐴𝐵 = ((1st ↾ (V × V)) “ 𝐴))
83, 5, 73bitr4i 306 1 (𝐴Domain𝐵𝐵 = dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1542  wcel 2114  Vcvv 3398   class class class wbr 5030   × cxp 5523  dom cdm 5525  cres 5527  cima 5528  1st c1st 7714  Imagecimage 33787  Domaincdomain 33790
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 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7481
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 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-symdif 4133  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-eprel 5434  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fo 6345  df-fv 6347  df-1st 7716  df-2nd 7717  df-txp 33801  df-image 33811  df-domain 33814
This theorem is referenced by:  brdomaing  33882  dfrecs2  33897  dfrdg4  33898
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