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Theorem brrangeg 33821
 Description: Closed form of brrange 33819. (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 5039 . . 3 (𝑎 = 𝐴 → (𝑎Range𝑏𝐴Range𝑏))
2 rneq 5782 . . . 4 (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴)
32eqeq2d 2769 . . 3 (𝑎 = 𝐴 → (𝑏 = ran 𝑎𝑏 = ran 𝐴))
41, 3bibi12d 349 . 2 (𝑎 = 𝐴 → ((𝑎Range𝑏𝑏 = ran 𝑎) ↔ (𝐴Range𝑏𝑏 = ran 𝐴)))
5 breq2 5040 . . 3 (𝑏 = 𝐵 → (𝐴Range𝑏𝐴Range𝐵))
6 eqeq1 2762 . . 3 (𝑏 = 𝐵 → (𝑏 = ran 𝐴𝐵 = ran 𝐴))
75, 6bibi12d 349 . 2 (𝑏 = 𝐵 → ((𝐴Range𝑏𝑏 = ran 𝐴) ↔ (𝐴Range𝐵𝐵 = ran 𝐴)))
8 vex 3413 . . 3 𝑎 ∈ V
9 vex 3413 . . 3 𝑏 ∈ V
108, 9brrange 33819 . 2 (𝑎Range𝑏𝑏 = ran 𝑎)
114, 7, 10vtocl2g 3492 1 ((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   class class class wbr 5036  ran crn 5529  Rangecrange 33729 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-symdif 4149  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-eprel 5439  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fo 6346  df-fv 6348  df-1st 7699  df-2nd 7700  df-txp 33739  df-image 33749  df-range 33753 This theorem is referenced by: (None)
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