Theorem brdomain 34162

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	⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)

Proof of Theorem brdomain

Step	Hyp	Ref	Expression
1		brdomain.1	. . 3 ⊢ 𝐴 ∈ V
2		brdomain.2	. . 3 ⊢ 𝐵 ∈ V
3	1, 2	brimage 34155	. 2 ⊢ (𝐴Image(1st ↾ (V × V))𝐵 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴))
4		df-domain 34096	. . 3 ⊢ Domain = Image(1st ↾ (V × V))
5	4	breqi 5076	. 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1st ↾ (V × V))𝐵)
6		dfdm5 33653	. . 3 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
7	6	eqeq2i 2751	. 2 ⊢ (𝐵 = dom 𝐴 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴))
8	3, 5, 7	3bitr4i 302	1 ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070   × cxp 5578  dom cdm 5580   ↾ cres 5582   “ cima 5583  1^st c1st 7802  Imagecimage 34069  Domaincdomain 34072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-symdif 4173  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-eprel 5486  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-1st 7804  df-2nd 7805  df-txp 34083  df-image 34093  df-domain 34096

This theorem is referenced by:  brdomaing  34164  dfrecs2  34179  dfrdg4  34180