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Theorem fnimapr 6740
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 6736 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
213adant3 1124 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
3 fnsnfv 6736 . . . . 5 ((𝐹 Fn 𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
433adant2 1123 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
52, 4uneq12d 4137 . . 3 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ({(𝐹𝐵)} ∪ {(𝐹𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
65eqcomd 2824 . 2 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐵)} ∪ {(𝐹𝐶)}))
7 df-pr 4560 . . . 4 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
87imaeq2i 5920 . . 3 (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9 imaundi 6001 . . 3 (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
108, 9eqtri 2841 . 2 (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11 df-pr 4560 . 2 {(𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐵)} ∪ {(𝐹𝐶)})
126, 10, 113eqtr4g 2878 1 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  cun 3931  {csn 4557  {cpr 4559  cima 5551   Fn wfn 6343  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  fvinim0ffz  13144  mrcun  16881  fnimatp  30351  s2rn  30547  poimirlem1  34774  poimirlem9  34782  imarnf1pr  43358  isomuspgrlem1  43869  isomuspgrlem2b  43871  isomuspgrlem2d  43873  isomuspgr  43876
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