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Theorem brimageg 35528
Description: Closed form of brimage 35527. (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 5153 . . 3 (𝑥 = 𝐴 → (𝑥Image𝑅𝑦𝐴Image𝑅𝑦))
2 imaeq2 6062 . . . 4 (𝑥 = 𝐴 → (𝑅𝑥) = (𝑅𝐴))
32eqeq2d 2738 . . 3 (𝑥 = 𝐴 → (𝑦 = (𝑅𝑥) ↔ 𝑦 = (𝑅𝐴)))
41, 3bibi12d 344 . 2 (𝑥 = 𝐴 → ((𝑥Image𝑅𝑦𝑦 = (𝑅𝑥)) ↔ (𝐴Image𝑅𝑦𝑦 = (𝑅𝐴))))
5 breq2 5154 . . 3 (𝑦 = 𝐵 → (𝐴Image𝑅𝑦𝐴Image𝑅𝐵))
6 eqeq1 2731 . . 3 (𝑦 = 𝐵 → (𝑦 = (𝑅𝐴) ↔ 𝐵 = (𝑅𝐴)))
75, 6bibi12d 344 . 2 (𝑦 = 𝐵 → ((𝐴Image𝑅𝑦𝑦 = (𝑅𝐴)) ↔ (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))))
8 vex 3475 . . 3 𝑥 ∈ V
9 vex 3475 . . 3 𝑦 ∈ V
108, 9brimage 35527 . 2 (𝑥Image𝑅𝑦𝑦 = (𝑅𝑥))
114, 7, 10vtocl2g 3560 1 ((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098   class class class wbr 5150  cima 5683  Imagecimage 35441
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-symdif 4243  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-eprel 5584  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fo 6557  df-fv 6559  df-1st 7997  df-2nd 7998  df-txp 35455  df-image 35465
This theorem is referenced by:  fnimage  35530
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