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Theorem brrangeg 32380
Description: Closed form of brrange 32378. (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 4789 . . 3 (𝑎 = 𝐴 → (𝑎Range𝑏𝐴Range𝑏))
2 rneq 5489 . . . 4 (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴)
32eqeq2d 2781 . . 3 (𝑎 = 𝐴 → (𝑏 = ran 𝑎𝑏 = ran 𝐴))
41, 3bibi12d 334 . 2 (𝑎 = 𝐴 → ((𝑎Range𝑏𝑏 = ran 𝑎) ↔ (𝐴Range𝑏𝑏 = ran 𝐴)))
5 breq2 4790 . . 3 (𝑏 = 𝐵 → (𝐴Range𝑏𝐴Range𝐵))
6 eqeq1 2775 . . 3 (𝑏 = 𝐵 → (𝑏 = ran 𝐴𝐵 = ran 𝐴))
75, 6bibi12d 334 . 2 (𝑏 = 𝐵 → ((𝐴Range𝑏𝑏 = ran 𝐴) ↔ (𝐴Range𝐵𝐵 = ran 𝐴)))
8 vex 3354 . . 3 𝑎 ∈ V
9 vex 3354 . . 3 𝑏 ∈ V
108, 9brrange 32378 . 2 (𝑎Range𝑏𝑏 = ran 𝑎)
114, 7, 10vtocl2g 3421 1 ((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145   class class class wbr 4786  ran crn 5250  Rangecrange 32288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-symdif 3993  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-eprel 5162  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7315  df-2nd 7316  df-txp 32298  df-image 32308  df-range 32312
This theorem is referenced by: (None)
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