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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 33285 | . 2 ⊢ (𝐴Image(2nd ↾ (V × V))𝐵 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴)) |
4 | df-range 33227 | . . 3 ⊢ Range = Image(2nd ↾ (V × V)) | |
5 | 4 | breqi 5064 | . 2 ⊢ (𝐴Range𝐵 ↔ 𝐴Image(2nd ↾ (V × V))𝐵) |
6 | dfrn5 32915 | . . 3 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴) | |
7 | 6 | eqeq2i 2834 | . 2 ⊢ (𝐵 = ran 𝐴 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴)) |
8 | 3, 5, 7 | 3bitr4i 304 | 1 ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 Vcvv 3495 class class class wbr 5058 × cxp 5547 ran crn 5550 ↾ cres 5551 “ cima 5552 2nd c2nd 7679 Imagecimage 33199 Rangecrange 33203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-symdif 4218 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-eprel 5459 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-1st 7680 df-2nd 7681 df-txp 33213 df-image 33223 df-range 33227 |
This theorem is referenced by: brrangeg 33295 brrestrict 33308 |
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