Theorem fnimapr 6992

Description: The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)

Assertion

Ref	Expression
fnimapr	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹‘𝐵), (𝐹‘𝐶)})

Proof of Theorem fnimapr

Step	Hyp	Ref	Expression
1		fnsnfv 6988	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
2	1	3adant3 1133	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
3		fnsnfv 6988	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
4	3	3adant2 1132	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
5	2, 4	uneq12d 4169	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
6	5	eqcomd 2743	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
7		df-pr 4629	. . . 4 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
8	7	imaeq2i 6076	. . 3 ⊢ (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9		imaundi 6169	. . 3 ⊢ (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
10	8, 9	eqtri 2765	. 2 ⊢ (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11		df-pr 4629	. 2 ⊢ {(𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
12	6, 10, 11	3eqtr4g 2802	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹‘𝐵), (𝐹‘𝐶)})

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ∪ cun 3949  {csn 4626  {cpr 4628   “ cima 5688   Fn wfn 6556  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is referenced by:  fnimatpd  6993  fvinim0ffz  13825  mrcun  17665  negs1s  28059  dfpth2  29749  s2rnOLD  32928  poimirlem1  37628  poimirlem9  37636  imarnf1pr  47294  uspgrimprop  47873  isuspgrimlem  47874  clnbgrgrimlem  47901  clnbgrgrim  47902  grimgrtri  47916  isubgr3stgrlem4  47936  isubgr3stgrlem7  47939  grlimgrtrilem2  47962