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Theorem brimage 35927
Description: Binary relation form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brimage.1 𝐴 ∈ V
brimage.2 𝐵 ∈ V
Assertion
Ref Expression
brimage (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))

Proof of Theorem brimage
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brimage.1 . 2 𝐴 ∈ V
2 brimage.2 . 2 𝐵 ∈ V
3 df-image 35865 . 2 Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝑅) ⊗ V)))
4 brxp 5734 . . 3 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
51, 2, 4mpbir2an 711 . 2 𝐴(V × V)𝐵
6 vex 3484 . . . . 5 𝑥 ∈ V
7 vex 3484 . . . . 5 𝑦 ∈ V
86, 7brcnv 5893 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
98rexbii 3094 . . 3 (∃𝑦𝐴 𝑥𝑅𝑦 ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
106, 1coep 35752 . . 3 (𝑥( E ∘ 𝑅)𝐴 ↔ ∃𝑦𝐴 𝑥𝑅𝑦)
116elima 6083 . . 3 (𝑥 ∈ (𝑅𝐴) ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
129, 10, 113bitr4ri 304 . 2 (𝑥 ∈ (𝑅𝐴) ↔ 𝑥( E ∘ 𝑅)𝐴)
131, 2, 3, 5, 12brtxpsd3 35897 1 (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480   class class class wbr 5143   E cep 5583   × cxp 5683  ccnv 5684  cima 5688  ccom 5689  Imagecimage 35841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-symdif 4253  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-image 35865
This theorem is referenced by:  brimageg  35928  funimage  35929  fnimage  35930  imageval  35931  brdomain  35934  brrange  35935  brimg  35938  funpartlem  35943  imagesset  35954
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