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Theorem brrangeg 31720
Description: Closed form of brrange 31718. (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 4621 . . 3 (𝑎 = 𝐴 → (𝑎Range𝑏𝐴Range𝑏))
2 rneq 5316 . . . 4 (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴)
32eqeq2d 2631 . . 3 (𝑎 = 𝐴 → (𝑏 = ran 𝑎𝑏 = ran 𝐴))
41, 3bibi12d 335 . 2 (𝑎 = 𝐴 → ((𝑎Range𝑏𝑏 = ran 𝑎) ↔ (𝐴Range𝑏𝑏 = ran 𝐴)))
5 breq2 4622 . . 3 (𝑏 = 𝐵 → (𝐴Range𝑏𝐴Range𝐵))
6 eqeq1 2625 . . 3 (𝑏 = 𝐵 → (𝑏 = ran 𝐴𝐵 = ran 𝐴))
75, 6bibi12d 335 . 2 (𝑏 = 𝐵 → ((𝐴Range𝑏𝑏 = ran 𝐴) ↔ (𝐴Range𝐵𝐵 = ran 𝐴)))
8 vex 3192 . . 3 𝑎 ∈ V
9 vex 3192 . . 3 𝑏 ∈ V
108, 9brrange 31718 . 2 (𝑎Range𝑏𝑏 = ran 𝑎)
114, 7, 10vtocl2g 3259 1 ((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987   class class class wbr 4618  ran crn 5080  Rangecrange 31627
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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-symdif 3827  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31637  df-image 31647  df-range 31651
This theorem is referenced by: (None)
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