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Theorem brimage 32908
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 32846 . 2 Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝑅) ⊗ V)))
4 brxp 5454 . . 3 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
51, 2, 4mpbir2an 698 . 2 𝐴(V × V)𝐵
6 vex 3418 . . . . 5 𝑥 ∈ V
7 vex 3418 . . . . 5 𝑦 ∈ V
86, 7brcnv 5604 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
98rexbii 3194 . . 3 (∃𝑦𝐴 𝑥𝑅𝑦 ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
106, 1coep 32507 . . 3 (𝑥( E ∘ 𝑅)𝐴 ↔ ∃𝑦𝐴 𝑥𝑅𝑦)
116elima 5777 . . 3 (𝑥 ∈ (𝑅𝐴) ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
129, 10, 113bitr4ri 296 . 2 (𝑥 ∈ (𝑅𝐴) ↔ 𝑥( E ∘ 𝑅)𝐴)
131, 2, 3, 5, 12brtxpsd3 32878 1 (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1507  wcel 2050  wrex 3089  Vcvv 3415   class class class wbr 4930   E cep 5317   × cxp 5406  ccnv 5407  cima 5411  ccom 5412  Imagecimage 32822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-symdif 4108  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-eprel 5318  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-fo 6196  df-fv 6198  df-1st 7503  df-2nd 7504  df-txp 32836  df-image 32846
This theorem is referenced by:  brimageg  32909  funimage  32910  fnimage  32911  imageval  32912  brdomain  32915  brrange  32916  brimg  32919  funpartlem  32924  imagesset  32935
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