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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 33389 | . 2 ⊢ (𝐴Image(2nd ↾ (V × V))𝐵 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴)) |
4 | df-range 33331 | . . 3 ⊢ Range = Image(2nd ↾ (V × V)) | |
5 | 4 | breqi 5074 | . 2 ⊢ (𝐴Range𝐵 ↔ 𝐴Image(2nd ↾ (V × V))𝐵) |
6 | dfrn5 33019 | . . 3 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴) | |
7 | 6 | eqeq2i 2836 | . 2 ⊢ (𝐵 = ran 𝐴 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴)) |
8 | 3, 5, 7 | 3bitr4i 305 | 1 ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3496 class class class wbr 5068 × cxp 5555 ran crn 5558 ↾ cres 5559 “ cima 5560 2nd c2nd 7690 Imagecimage 33303 Rangecrange 33307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-symdif 4221 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-eprel 5467 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-1st 7691 df-2nd 7692 df-txp 33317 df-image 33327 df-range 33331 |
This theorem is referenced by: brrangeg 33399 brrestrict 33412 |
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