Theorem brdomain 35915

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 35908	. 2 ⊢ (𝐴Image(1st ↾ (V × V))𝐵 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴))
4		df-domain 35849	. . 3 ⊢ Domain = Image(1st ↾ (V × V))
5	4	breqi 5154	. 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1st ↾ (V × V))𝐵)
6		dfdm5 35754	. . 3 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
7	6	eqeq2i 2748	. 2 ⊢ (𝐵 = dom 𝐴 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴))
8	3, 5, 7	3bitr4i 303	1 ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2106  Vcvv 3478   class class class wbr 5148   × cxp 5687  dom cdm 5689   ↾ cres 5691   “ cima 5692  1^st c1st 8011  Imagecimage 35822  Domaincdomain 35825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-image 35846  df-domain 35849

This theorem is referenced by:  brdomaing  35917  dfrecs2  35932  dfrdg4  35933