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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 35555 | . 2 ⊢ (𝐴Image(1st ↾ (V × V))𝐵 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴)) |
4 | df-domain 35496 | . . 3 ⊢ Domain = Image(1st ↾ (V × V)) | |
5 | 4 | breqi 5158 | . 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1st ↾ (V × V))𝐵) |
6 | dfdm5 35401 | . . 3 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴) | |
7 | 6 | eqeq2i 2741 | . 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 1533 ∈ wcel 2098 Vcvv 3473 class class class wbr 5152 × cxp 5680 dom cdm 5682 ↾ cres 5684 “ cima 5685 1st c1st 7997 Imagecimage 35469 Domaincdomain 35472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-symdif 4245 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-eprel 5586 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-fv 6561 df-1st 7999 df-2nd 8000 df-txp 35483 df-image 35493 df-domain 35496 |
This theorem is referenced by: brdomaing 35564 dfrecs2 35579 dfrdg4 35580 |
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