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Theorem brimageg 31676
Description: Closed form of brimage 31675. (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 4616 . . 3 (𝑥 = 𝐴 → (𝑥Image𝑅𝑦𝐴Image𝑅𝑦))
2 imaeq2 5421 . . . 4 (𝑥 = 𝐴 → (𝑅𝑥) = (𝑅𝐴))
32eqeq2d 2631 . . 3 (𝑥 = 𝐴 → (𝑦 = (𝑅𝑥) ↔ 𝑦 = (𝑅𝐴)))
41, 3bibi12d 335 . 2 (𝑥 = 𝐴 → ((𝑥Image𝑅𝑦𝑦 = (𝑅𝑥)) ↔ (𝐴Image𝑅𝑦𝑦 = (𝑅𝐴))))
5 breq2 4617 . . 3 (𝑦 = 𝐵 → (𝐴Image𝑅𝑦𝐴Image𝑅𝐵))
6 eqeq1 2625 . . 3 (𝑦 = 𝐵 → (𝑦 = (𝑅𝐴) ↔ 𝐵 = (𝑅𝐴)))
75, 6bibi12d 335 . 2 (𝑦 = 𝐵 → ((𝐴Image𝑅𝑦𝑦 = (𝑅𝐴)) ↔ (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))))
8 vex 3189 . . 3 𝑥 ∈ V
9 vex 3189 . . 3 𝑦 ∈ V
108, 9brimage 31675 . 2 (𝑥Image𝑅𝑦𝑦 = (𝑅𝑥))
114, 7, 10vtocl2g 3256 1 ((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987   class class class wbr 4613  cima 5077  Imagecimage 31588
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-symdif 3822  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-eprel 4985  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113  df-2nd 7114  df-txp 31602  df-image 31612
This theorem is referenced by:  fnimage  31678
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