Theorem brdomain 35921

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 35914	. 2 ⊢ (𝐴Image(1st ↾ (V × V))𝐵 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴))
4		df-domain 35855	. . 3 ⊢ Domain = Image(1st ↾ (V × V))
5	4	breqi 5113	. 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1st ↾ (V × V))𝐵)
6		dfdm5 35760	. . 3 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
7	6	eqeq2i 2742	. 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 1540   ∈ wcel 2109  Vcvv 3447   class class class wbr 5107   × cxp 5636  dom cdm 5638   ↾ cres 5640   “ cima 5641  1^st c1st 7966  Imagecimage 35828  Domaincdomain 35831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-symdif 4216  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-eprel 5538  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-1st 7968  df-2nd 7969  df-txp 35842  df-image 35852  df-domain 35855

This theorem is referenced by:  brdomaing  35923  dfrecs2  35938  dfrdg4  35939