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Theorem fnimapr 5374
 Description: The image of a pair under a funtion. (Contributed by Jeff Madsen, 6-Jan-2011.)
Assertion
Ref Expression
fnimapr ((F Fn A B A C A) → (F “ {B, C}) = {(FB), (FC)})

Proof of Theorem fnimapr
StepHypRef Expression
1 fnsnfv 5373 . . . . 5 ((F Fn A B A) → {(FB)} = (F “ {B}))
213adant3 975 . . . 4 ((F Fn A B A C A) → {(FB)} = (F “ {B}))
3 fnsnfv 5373 . . . . 5 ((F Fn A C A) → {(FC)} = (F “ {C}))
433adant2 974 . . . 4 ((F Fn A B A C A) → {(FC)} = (F “ {C}))
52, 4uneq12d 3419 . . 3 ((F Fn A B A C A) → ({(FB)} ∪ {(FC)}) = ((F “ {B}) ∪ (F “ {C})))
65eqcomd 2358 . 2 ((F Fn A B A C A) → ((F “ {B}) ∪ (F “ {C})) = ({(FB)} ∪ {(FC)}))
7 df-pr 3742 . . . 4 {B, C} = ({B} ∪ {C})
87imaeq2i 4940 . . 3 (F “ {B, C}) = (F “ ({B} ∪ {C}))
9 imaundi 5039 . . 3 (F “ ({B} ∪ {C})) = ((F “ {B}) ∪ (F “ {C}))
108, 9eqtri 2373 . 2 (F “ {B, C}) = ((F “ {B}) ∪ (F “ {C}))
11 df-pr 3742 . 2 {(FB), (FC)} = ({(FB)} ∪ {(FC)})
126, 10, 113eqtr4g 2410 1 ((F Fn A B A C A) → (F “ {B, C}) = {(FB), (FC)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ∪ cun 3207  {csn 3737  {cpr 3738   “ cima 4722   Fn wfn 4776   ‘cfv 4781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795 This theorem is referenced by: (None)
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