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Theorem fnimapr 5589
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 5588 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
213adant3 1018 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
3 fnsnfv 5588 . . . . 5 ((𝐹 Fn 𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
433adant2 1017 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
52, 4uneq12d 3302 . . 3 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ({(𝐹𝐵)} ∪ {(𝐹𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
65eqcomd 2193 . 2 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐵)} ∪ {(𝐹𝐶)}))
7 df-pr 3611 . . . 4 {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
87imaeq2i 4980 . . 3 (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9 imaundi 5053 . . 3 (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
108, 9eqtri 2208 . 2 (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11 df-pr 3611 . 2 {(𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐵)} ∪ {(𝐹𝐶)})
126, 10, 113eqtr4g 2245 1 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 979   = wceq 1363  wcel 2158  cun 3139  {csn 3604  {cpr 3605  cima 4641   Fn wfn 5223  cfv 5228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236
This theorem is referenced by:  fvinim0ffz  10255
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