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Theorem brimage 32017
Description: Binary relationship 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 31955 . 2 Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝑅) ⊗ V)))
4 brxp 5145 . . 3 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
51, 2, 4mpbir2an 955 . 2 𝐴(V × V)𝐵
6 vex 3201 . . . . 5 𝑥 ∈ V
7 vex 3201 . . . . 5 𝑦 ∈ V
86, 7brcnv 5303 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
98rexbii 3039 . . 3 (∃𝑦𝐴 𝑥𝑅𝑦 ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
106, 1coep 31627 . . 3 (𝑥( E ∘ 𝑅)𝐴 ↔ ∃𝑦𝐴 𝑥𝑅𝑦)
116elima 5469 . . 3 (𝑥 ∈ (𝑅𝐴) ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
129, 10, 113bitr4ri 293 . 2 (𝑥 ∈ (𝑅𝐴) ↔ 𝑥( E ∘ 𝑅)𝐴)
131, 2, 3, 5, 12brtxpsd3 31987 1 (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1482  wcel 1989  wrex 2912  Vcvv 3198   class class class wbr 4651   E cep 5026   × cxp 5110  ccnv 5111  cima 5115  ccom 5116  Imagecimage 31931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-symdif 3842  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-eprel 5027  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fo 5892  df-fv 5894  df-1st 7165  df-2nd 7166  df-txp 31945  df-image 31955
This theorem is referenced by:  brimageg  32018  funimage  32019  fnimage  32020  imageval  32021  brdomain  32024  brrange  32025  brimg  32028  funpartlem  32033  imagesset  32044
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