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Theorem brdomain 32015
Description: The binary relationship 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 32008 . 2 (𝐴Image(1st ↾ (V × V))𝐵𝐵 = ((1st ↾ (V × V)) “ 𝐴))
4 df-domain 31948 . . 3 Domain = Image(1st ↾ (V × V))
54breqi 4650 . 2 (𝐴Domain𝐵𝐴Image(1st ↾ (V × V))𝐵)
6 dfdm5 31650 . . 3 dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
76eqeq2i 2632 . 2 (𝐵 = dom 𝐴𝐵 = ((1st ↾ (V × V)) “ 𝐴))
83, 5, 73bitr4i 292 1 (𝐴Domain𝐵𝐵 = dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1481  wcel 1988  Vcvv 3195   class class class wbr 4644   × cxp 5102  dom cdm 5104  cres 5106  cima 5107  1st c1st 7151  Imagecimage 31921  Domaincdomain 31924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-symdif 3836  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-eprel 5019  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-1st 7153  df-2nd 7154  df-txp 31935  df-image 31945  df-domain 31948
This theorem is referenced by:  brdomaing  32017  dfrecs2  32032  dfrdg4  32033
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