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Mirrors > Home > MPE Home > Th. List > Mathboxes > brrange | Structured version Visualization version GIF version |
Description: Binary relation form of the range 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 |
---|---|
brrange | ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomain.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | brdomain.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | brimage 35527 | . 2 ⊢ (𝐴Image(2nd ↾ (V × V))𝐵 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴)) |
4 | df-range 35469 | . . 3 ⊢ Range = Image(2nd ↾ (V × V)) | |
5 | 4 | breqi 5156 | . 2 ⊢ (𝐴Range𝐵 ↔ 𝐴Image(2nd ↾ (V × V))𝐵) |
6 | dfrn5 35374 | . . 3 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴) | |
7 | 6 | eqeq2i 2740 | . 2 ⊢ (𝐵 = ran 𝐴 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴)) |
8 | 3, 5, 7 | 3bitr4i 302 | 1 ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 Vcvv 3471 class class class wbr 5150 × cxp 5678 ran crn 5681 ↾ cres 5682 “ cima 5683 2nd c2nd 7996 Imagecimage 35441 Rangecrange 35445 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-symdif 4243 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-eprel 5584 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-fo 6557 df-fv 6559 df-1st 7997 df-2nd 7998 df-txp 35455 df-image 35465 df-range 35469 |
This theorem is referenced by: brrangeg 35537 brrestrict 35550 |
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