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Theorem brrangeg 35403
Description: Closed form of brrange 35401. (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 5141 . . 3 (𝑎 = 𝐴 → (𝑎Range𝑏𝐴Range𝑏))
2 rneq 5925 . . . 4 (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴)
32eqeq2d 2735 . . 3 (𝑎 = 𝐴 → (𝑏 = ran 𝑎𝑏 = ran 𝐴))
41, 3bibi12d 345 . 2 (𝑎 = 𝐴 → ((𝑎Range𝑏𝑏 = ran 𝑎) ↔ (𝐴Range𝑏𝑏 = ran 𝐴)))
5 breq2 5142 . . 3 (𝑏 = 𝐵 → (𝐴Range𝑏𝐴Range𝐵))
6 eqeq1 2728 . . 3 (𝑏 = 𝐵 → (𝑏 = ran 𝐴𝐵 = ran 𝐴))
75, 6bibi12d 345 . 2 (𝑏 = 𝐵 → ((𝐴Range𝑏𝑏 = ran 𝐴) ↔ (𝐴Range𝐵𝐵 = ran 𝐴)))
8 vex 3470 . . 3 𝑎 ∈ V
9 vex 3470 . . 3 𝑏 ∈ V
108, 9brrange 35401 . 2 (𝑎Range𝑏𝑏 = ran 𝑎)
114, 7, 10vtocl2g 3555 1 ((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098   class class class wbr 5138  ran crn 5667  Rangecrange 35311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-symdif 4234  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-eprel 5570  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-1st 7968  df-2nd 7969  df-txp 35321  df-image 35331  df-range 35335
This theorem is referenced by: (None)
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