brdomain - Mathbox for Scott Fenton

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Theorem brdomain 32165

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 32158	. 2 ⊢ (𝐴Image(1^st ↾ (V × V))𝐵 ↔ 𝐵 = ((1^st ↾ (V × V)) “ 𝐴))
4		df-domain 32099	. . 3 ⊢ Domain = Image(1^st ↾ (V × V))
5	4	breqi 4691	. 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1^st ↾ (V × V))𝐵)
6		dfdm5 31800	. . 3 ⊢ dom 𝐴 = ((1^st ↾ (V × V)) “ 𝐴)
7	6	eqeq2i 2663	. 2 ⊢ (𝐵 = dom 𝐴 ↔ 𝐵 = ((1^st ↾ (V × V)) “ 𝐴))
8	3, 5, 7	3bitr4i 292	1 ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 = wceq 1523 ∈ wcel 2030 Vcvv 3231 class class class wbr 4685 × cxp 5141 dom cdm 5143 ↾ cres 5145 “ cima 5146 1^st c1st 7208 Imagecimage 32072 Domaincdomain 32075

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-symdif 3877 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-eprel 5058 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fo 5932 df-fv 5934 df-1st 7210 df-2nd 7211 df-txp 32086 df-image 32096 df-domain 32099

This theorem is referenced by: brdomaing 32167 dfrecs2 32182 dfrdg4 32183

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