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Theorem brrange 34281
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 34273 . 2 (𝐴Image(2nd ↾ (V × V))𝐵𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
4 df-range 34215 . . 3 Range = Image(2nd ↾ (V × V))
54breqi 5087 . 2 (𝐴Range𝐵𝐴Image(2nd ↾ (V × V))𝐵)
6 dfrn5 33793 . . 3 ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
76eqeq2i 2749 . 2 (𝐵 = ran 𝐴𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
83, 5, 73bitr4i 303 1 (𝐴Range𝐵𝐵 = ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  wcel 2104  Vcvv 3437   class class class wbr 5081   × cxp 5598  ran crn 5601  cres 5602  cima 5603  2nd c2nd 7862  Imagecimage 34187  Rangecrange 34191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-symdif 4182  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-eprel 5506  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fo 6464  df-fv 6466  df-1st 7863  df-2nd 7864  df-txp 34201  df-image 34211  df-range 34215
This theorem is referenced by:  brrangeg  34283  brrestrict  34296
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