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Theorem fnimapr 6992
Description: The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
Assertion
Ref Expression
fnimapr ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})

Proof of Theorem fnimapr
StepHypRef Expression
1 fnsnfv 6988 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
213adant3 1131 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
3 fnsnfv 6988 . . . . 5 ((𝐹 Fn 𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
433adant2 1130 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
52, 4uneq12d 4179 . . 3 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ({(𝐹𝐵)} ∪ {(𝐹𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
65eqcomd 2741 . 2 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐵)} ∪ {(𝐹𝐶)}))
7 df-pr 4634 . . . 4 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
87imaeq2i 6078 . . 3 (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9 imaundi 6172 . . 3 (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
108, 9eqtri 2763 . 2 (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11 df-pr 4634 . 2 {(𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐵)} ∪ {(𝐹𝐶)})
126, 10, 113eqtr4g 2800 1 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  cun 3961  {csn 4631  {cpr 4633  cima 5692   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  fnimatpd  6993  fvinim0ffz  13822  mrcun  17667  negs1s  28074  s2rnOLD  32913  poimirlem1  37608  poimirlem9  37616  imarnf1pr  47232  uspgrimprop  47811  isuspgrimlem  47812  clnbgrgrimlem  47839  clnbgrgrim  47840  grimgrtri  47852  isubgr3stgrlem4  47872  isubgr3stgrlem7  47875  grlimgrtrilem2  47898
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