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Theorem brdomain 32544
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 32537 . 2 (𝐴Image(1st ↾ (V × V))𝐵𝐵 = ((1st ↾ (V × V)) “ 𝐴))
4 df-domain 32478 . . 3 Domain = Image(1st ↾ (V × V))
54breqi 4847 . 2 (𝐴Domain𝐵𝐴Image(1st ↾ (V × V))𝐵)
6 dfdm5 32179 . . 3 dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
76eqeq2i 2809 . 2 (𝐵 = dom 𝐴𝐵 = ((1st ↾ (V × V)) “ 𝐴))
83, 5, 73bitr4i 295 1 (𝐴Domain𝐵𝐵 = dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  wcel 2157  Vcvv 3383   class class class wbr 4841   × cxp 5308  dom cdm 5310  cres 5312  cima 5313  1st c1st 7397  Imagecimage 32451  Domaincdomain 32454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-symdif 4039  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-eprel 5223  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fo 6105  df-fv 6107  df-1st 7399  df-2nd 7400  df-txp 32465  df-image 32475  df-domain 32478
This theorem is referenced by:  brdomaing  32546  dfrecs2  32561  dfrdg4  32562
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