Theorem brdomain 35951

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 35944	. 2 ⊢ (𝐴Image(1st ↾ (V × V))𝐵 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴))
4		df-domain 35885	. . 3 ⊢ Domain = Image(1st ↾ (V × V))
5	4	breqi 5125	. 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1st ↾ (V × V))𝐵)
6		dfdm5 35790	. . 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 1540   ∈ wcel 2108  Vcvv 3459   class class class wbr 5119   × cxp 5652  dom cdm 5654   ↾ cres 5656   “ cima 5657  1^st c1st 7986  Imagecimage 35858  Domaincdomain 35861

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-symdif 4228  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-eprel 5553  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-1st 7988  df-2nd 7989  df-txp 35872  df-image 35882  df-domain 35885

This theorem is referenced by:  brdomaing  35953  dfrecs2  35968  dfrdg4  35969