Theorem fnimapr 6935

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 6931	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
2	1	3adant3 1141	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
3		fnsnfv 6931	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
4	3	3adant2 1140	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
5	2, 4	uneq12d 4113	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
6	5	eqcomd 2758	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
7		df-pr 4575	. . . 4 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
8	7	imaeq2i 6033	. . 3 ⊢ (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9		imaundi 6120	. . 3 ⊢ (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
10	8, 9	eqtri 2775	. 2 ⊢ (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11		df-pr 4575	. 2 ⊢ {(𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
12	6, 10, 11	3eqtr4g 2812	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹‘𝐵), (𝐹‘𝐶)})

Syntax hints:   → wi 4   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ∪ cun 3893  {csn 4572  {cpr 4574   “ cima 5639   Fn wfn 6501  ‘cfv 6506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-fv 6514

This theorem is referenced by:  fnimatpd  6936  fvinim0ffz  13781  mrcun  17626  neg1s  28086  dfpth2  29864  s2rnOLD  33072  poimirlem1  38058  poimirlem9  38066  imarnf1pr  47814  uhgrimprop  48452  isuspgrimlem  48455  upgrimwlklem5  48461  upgrimpths  48469  clnbgrgrimlem  48493  clnbgrgrim  48494  grimgrtri  48509  isubgr3stgrlem4  48529  isubgr3stgrlem7  48532  grlimprclnbgr  48556  grlimgrtrilem2  48562