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Theorem brrange 33395
Description: Binary relation form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brdomain.1 𝐴 ∈ V
brdomain.2 𝐵 ∈ V
Assertion
Ref Expression
brrange (𝐴Range𝐵𝐵 = ran 𝐴)

Proof of Theorem brrange
StepHypRef Expression
1 brdomain.1 . . 3 𝐴 ∈ V
2 brdomain.2 . . 3 𝐵 ∈ V
31, 2brimage 33387 . 2 (𝐴Image(2nd ↾ (V × V))𝐵𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
4 df-range 33329 . . 3 Range = Image(2nd ↾ (V × V))
54breqi 5071 . 2 (𝐴Range𝐵𝐴Image(2nd ↾ (V × V))𝐵)
6 dfrn5 33017 . . 3 ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
76eqeq2i 2834 . 2 (𝐵 = ran 𝐴𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
83, 5, 73bitr4i 305 1 (𝐴Range𝐵𝐵 = ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 208   = wceq 1533  wcel 2110  Vcvv 3494   class class class wbr 5065   × cxp 5552  ran crn 5555  cres 5556  cima 5557  2nd c2nd 7687  Imagecimage 33301  Rangecrange 33305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  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 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-symdif 4218  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-eprel 5464  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-1st 7688  df-2nd 7689  df-txp 33315  df-image 33325  df-range 33329
This theorem is referenced by:  brrangeg  33397  brrestrict  33410
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