![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
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 35763 | . 2 ⊢ (𝐴Image(1st ↾ (V × V))𝐵 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴)) |
4 | df-domain 35704 | . . 3 ⊢ Domain = Image(1st ↾ (V × V)) | |
5 | 4 | breqi 5151 | . 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1st ↾ (V × V))𝐵) |
6 | dfdm5 35609 | . . 3 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴) | |
7 | 6 | eqeq2i 2739 | . 2 ⊢ (𝐵 = dom 𝐴 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴)) |
8 | 3, 5, 7 | 3bitr4i 302 | 1 ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 Vcvv 3462 class class class wbr 5145 × cxp 5672 dom cdm 5674 ↾ cres 5676 “ cima 5677 1st c1st 7993 Imagecimage 35677 Domaincdomain 35680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-symdif 4241 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-eprel 5578 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fo 6552 df-fv 6554 df-1st 7995 df-2nd 7996 df-txp 35691 df-image 35701 df-domain 35704 |
This theorem is referenced by: brdomaing 35772 dfrecs2 35787 dfrdg4 35788 |
Copyright terms: Public domain | W3C validator |