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Theorem fnimapr 6510
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 6506 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
213adant3 1168 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
3 fnsnfv 6506 . . . . 5 ((𝐹 Fn 𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
433adant2 1167 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
52, 4uneq12d 3996 . . 3 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ({(𝐹𝐵)} ∪ {(𝐹𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
65eqcomd 2832 . 2 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐵)} ∪ {(𝐹𝐶)}))
7 df-pr 4401 . . . 4 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
87imaeq2i 5706 . . 3 (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9 imaundi 5787 . . 3 (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
108, 9eqtri 2850 . 2 (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11 df-pr 4401 . 2 {(𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐵)} ∪ {(𝐹𝐶)})
126, 10, 113eqtr4g 2887 1 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1113   = wceq 1658  wcel 2166  cun 3797  {csn 4398  {cpr 4400  cima 5346   Fn wfn 6119  cfv 6124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-fv 6132
This theorem is referenced by:  fvinim0ffz  12883  mrcun  16636  poimirlem1  33955  poimirlem9  33963  imarnf1pr  42186  isomuspgrlem1  42546  isomuspgrlem2b  42548  isomuspgrlem2d  42550  isomuspgr  42553
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