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Theorem brrange 32554
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 32546 . 2 (𝐴Image(2nd ↾ (V × V))𝐵𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
4 df-range 32488 . . 3 Range = Image(2nd ↾ (V × V))
54breqi 4849 . 2 (𝐴Range𝐵𝐴Image(2nd ↾ (V × V))𝐵)
6 dfrn5 32189 . . 3 ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
76eqeq2i 2811 . 2 (𝐵 = ran 𝐴𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
83, 5, 73bitr4i 295 1 (𝐴Range𝐵𝐵 = ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  wcel 2157  Vcvv 3385   class class class wbr 4843   × cxp 5310  ran crn 5313  cres 5314  cima 5315  2nd c2nd 7400  Imagecimage 32460  Rangecrange 32464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-symdif 4041  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-eprel 5225  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7401  df-2nd 7402  df-txp 32474  df-image 32484  df-range 32488
This theorem is referenced by:  brrangeg  32556  brrestrict  32569
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