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Theorem brimage 32331
 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 32269 . 2 Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝑅) ⊗ V)))
4 brxp 5296 . . 3 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
51, 2, 4mpbir2an 993 . 2 𝐴(V × V)𝐵
6 vex 3335 . . . . 5 𝑥 ∈ V
7 vex 3335 . . . . 5 𝑦 ∈ V
86, 7brcnv 5452 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
98rexbii 3171 . . 3 (∃𝑦𝐴 𝑥𝑅𝑦 ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
106, 1coep 31940 . . 3 (𝑥( E ∘ 𝑅)𝐴 ↔ ∃𝑦𝐴 𝑥𝑅𝑦)
116elima 5621 . . 3 (𝑥 ∈ (𝑅𝐴) ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
129, 10, 113bitr4ri 293 . 2 (𝑥 ∈ (𝑅𝐴) ↔ 𝑥( E ∘ 𝑅)𝐴)
131, 2, 3, 5, 12brtxpsd3 32301 1 (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1624   ∈ wcel 2131  ∃wrex 3043  Vcvv 3332   class class class wbr 4796   E cep 5170   × cxp 5256  ◡ccnv 5257   “ cima 5261   ∘ ccom 5262  Imagecimage 32245 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-symdif 3979  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-eprel 5171  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fo 6047  df-fv 6049  df-1st 7325  df-2nd 7326  df-txp 32259  df-image 32269 This theorem is referenced by:  brimageg  32332  funimage  32333  fnimage  32334  imageval  32335  brdomain  32338  brrange  32339  brimg  32342  funpartlem  32347  imagesset  32358
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