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Theorem brimageg 34622
Description: Closed form of brimage 34621. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
brimageg ((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))

Proof of Theorem brimageg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5128 . . 3 (𝑥 = 𝐴 → (𝑥Image𝑅𝑦𝐴Image𝑅𝑦))
2 imaeq2 6029 . . . 4 (𝑥 = 𝐴 → (𝑅𝑥) = (𝑅𝐴))
32eqeq2d 2742 . . 3 (𝑥 = 𝐴 → (𝑦 = (𝑅𝑥) ↔ 𝑦 = (𝑅𝐴)))
41, 3bibi12d 345 . 2 (𝑥 = 𝐴 → ((𝑥Image𝑅𝑦𝑦 = (𝑅𝑥)) ↔ (𝐴Image𝑅𝑦𝑦 = (𝑅𝐴))))
5 breq2 5129 . . 3 (𝑦 = 𝐵 → (𝐴Image𝑅𝑦𝐴Image𝑅𝐵))
6 eqeq1 2735 . . 3 (𝑦 = 𝐵 → (𝑦 = (𝑅𝐴) ↔ 𝐵 = (𝑅𝐴)))
75, 6bibi12d 345 . 2 (𝑦 = 𝐵 → ((𝐴Image𝑅𝑦𝑦 = (𝑅𝐴)) ↔ (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))))
8 vex 3463 . . 3 𝑥 ∈ V
9 vex 3463 . . 3 𝑦 ∈ V
108, 9brimage 34621 . 2 (𝑥Image𝑅𝑦𝑦 = (𝑅𝑥))
114, 7, 10vtocl2g 3545 1 ((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106   class class class wbr 5125  cima 5656  Imagecimage 34535
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-symdif 4222  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-eprel 5557  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-fo 6522  df-fv 6524  df-1st 7941  df-2nd 7942  df-txp 34549  df-image 34559
This theorem is referenced by:  fnimage  34624
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