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Theorem brrangeg 32490
Description: Closed form of brrange 32488. (Contributed by Scott Fenton, 3-May-2014.)
Assertion
Ref Expression
brrangeg ((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))

Proof of Theorem brrangeg
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4814 . . 3 (𝑎 = 𝐴 → (𝑎Range𝑏𝐴Range𝑏))
2 rneq 5521 . . . 4 (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴)
32eqeq2d 2775 . . 3 (𝑎 = 𝐴 → (𝑏 = ran 𝑎𝑏 = ran 𝐴))
41, 3bibi12d 336 . 2 (𝑎 = 𝐴 → ((𝑎Range𝑏𝑏 = ran 𝑎) ↔ (𝐴Range𝑏𝑏 = ran 𝐴)))
5 breq2 4815 . . 3 (𝑏 = 𝐵 → (𝐴Range𝑏𝐴Range𝐵))
6 eqeq1 2769 . . 3 (𝑏 = 𝐵 → (𝑏 = ran 𝐴𝐵 = ran 𝐴))
75, 6bibi12d 336 . 2 (𝑏 = 𝐵 → ((𝐴Range𝑏𝑏 = ran 𝐴) ↔ (𝐴Range𝐵𝐵 = ran 𝐴)))
8 vex 3353 . . 3 𝑎 ∈ V
9 vex 3353 . . 3 𝑏 ∈ V
108, 9brrange 32488 . 2 (𝑎Range𝑏𝑏 = ran 𝑎)
114, 7, 10vtocl2g 3422 1 ((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155   class class class wbr 4811  ran crn 5280  Rangecrange 32398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-symdif 4007  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-eprel 5192  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fo 6076  df-fv 6078  df-1st 7368  df-2nd 7369  df-txp 32408  df-image 32418  df-range 32422
This theorem is referenced by: (None)
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