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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 33873 | . 2 ⊢ (𝐴Image(1st ↾ (V × V))𝐵 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴)) |
4 | df-domain 33814 | . . 3 ⊢ Domain = Image(1st ↾ (V × V)) | |
5 | 4 | breqi 5036 | . 2 ⊢ (𝐴Domain𝐵 ↔ 𝐴Image(1st ↾ (V × V))𝐵) |
6 | dfdm5 33323 | . . 3 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴) | |
7 | 6 | eqeq2i 2751 | . 2 ⊢ (𝐵 = dom 𝐴 ↔ 𝐵 = ((1st ↾ (V × V)) “ 𝐴)) |
8 | 3, 5, 7 | 3bitr4i 306 | 1 ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 ∈ wcel 2114 Vcvv 3398 class class class wbr 5030 × cxp 5523 dom cdm 5525 ↾ cres 5527 “ cima 5528 1st c1st 7714 Imagecimage 33787 Domaincdomain 33790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7481 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-symdif 4133 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-eprel 5434 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fo 6345 df-fv 6347 df-1st 7716 df-2nd 7717 df-txp 33801 df-image 33811 df-domain 33814 |
This theorem is referenced by: brdomaing 33882 dfrecs2 33897 dfrdg4 33898 |
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