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Theorem fnimapr 6219
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 6215 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
213adant3 1079 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
3 fnsnfv 6215 . . . . 5 ((𝐹 Fn 𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
433adant2 1078 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
52, 4uneq12d 3746 . . 3 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ({(𝐹𝐵)} ∪ {(𝐹𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
65eqcomd 2627 . 2 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐵)} ∪ {(𝐹𝐶)}))
7 df-pr 4151 . . . 4 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
87imaeq2i 5423 . . 3 (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9 imaundi 5504 . . 3 (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
108, 9eqtri 2643 . 2 (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11 df-pr 4151 . 2 {(𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐵)} ∪ {(𝐹𝐶)})
126, 10, 113eqtr4g 2680 1 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  cun 3553  {csn 4148  {cpr 4150  cima 5077   Fn wfn 5842  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855
This theorem is referenced by:  fvinim0ffz  12527  mrcun  16203  poimirlem1  33042  poimirlem9  33050  imarnf1pr  40598
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