brdomain - Mathbox for Scott Fenton

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Theorem brdomain 32544

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 32537	. 2 ⊢ (𝐴Image(1^st ↾ (V × V))𝐵 ↔ 𝐵 = ((1^st ↾ (V × V)) “ 𝐴))
4		df-domain 32478	. . 3 ⊢ Domain = Image(1^st ↾ (V × V))
5	4	breqi 4847	. 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1^st ↾ (V × V))𝐵)
6		dfdm5 32179	. . 3 ⊢ dom 𝐴 = ((1^st ↾ (V × V)) “ 𝐴)
7	6	eqeq2i 2809	. 2 ⊢ (𝐵 = dom 𝐴 ↔ 𝐵 = ((1^st ↾ (V × V)) “ 𝐴))
8	3, 5, 7	3bitr4i 295	1 ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 198 = wceq 1653 ∈ wcel 2157 Vcvv 3383 class class class wbr 4841 × cxp 5308 dom cdm 5310 ↾ cres 5312 “ cima 5313 1^st c1st 7397 Imagecimage 32451 Domaincdomain 32454

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-symdif 4039 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-eprel 5223 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-fo 6105 df-fv 6107 df-1st 7399 df-2nd 7400 df-txp 32465 df-image 32475 df-domain 32478

This theorem is referenced by: brdomaing 32546 dfrecs2 32561 dfrdg4 32562

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