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Theorem brrange 31736
Description: The binary relationship 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 31728 . 2 (𝐴Image(2nd ↾ (V × V))𝐵𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
4 df-range 31669 . . 3 Range = Image(2nd ↾ (V × V))
54breqi 4629 . 2 (𝐴Range𝐵𝐴Image(2nd ↾ (V × V))𝐵)
6 dfrn5 31432 . . 3 ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
76eqeq2i 2633 . 2 (𝐵 = ran 𝐴𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
83, 5, 73bitr4i 292 1 (𝐴Range𝐵𝐵 = ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  Vcvv 3190   class class class wbr 4623   × cxp 5082  ran crn 5085  cres 5086  cima 5087  2nd c2nd 7127  Imagecimage 31641  Rangecrange 31645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-symdif 3828  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-eprel 4995  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-1st 7128  df-2nd 7129  df-txp 31655  df-image 31665  df-range 31669
This theorem is referenced by:  brrangeg  31738  brrestrict  31751
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