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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brdomain | Structured version Visualization version GIF version | ||
| Description: Binary relation form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| Ref | Expression |
|---|---|
| brdomain.1 | ⊢ 𝐴 ∈ V |
| brdomain.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| brdomain | ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brdomain.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 2 | brdomain.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 3 | 1, 2 | brimage 35927 | . 2 ⊢ (𝐴Image(1st ↾ (V × V))𝐵 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴)) |
| 4 | df-domain 35868 | . . 3 ⊢ Domain = Image(1st ↾ (V × V)) | |
| 5 | 4 | breqi 5149 | . 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1st ↾ (V × V))𝐵) |
| 6 | dfdm5 35773 | . . 3 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴) | |
| 7 | 6 | eqeq2i 2750 | . 2 ⊢ (𝐵 = dom 𝐴 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴)) |
| 8 | 3, 5, 7 | 3bitr4i 303 | 1 ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 × cxp 5683 dom cdm 5685 ↾ cres 5687 “ cima 5688 1st c1st 8012 Imagecimage 35841 Domaincdomain 35844 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-symdif 4253 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 df-image 35865 df-domain 35868 |
| This theorem is referenced by: brdomaing 35936 dfrecs2 35951 dfrdg4 35952 |
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