Theorem fnimapr 6964

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 6960	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
2	1	3adant3 1149	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
3		fnsnfv 6960	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
4	3	3adant2 1148	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
5	2, 4	uneq12d 4122	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
6	5	eqcomd 2768	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
7		df-pr 4591	. . . 4 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
8	7	imaeq2i 6059	. . 3 ⊢ (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9		imaundi 6146	. . 3 ⊢ (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
10	8, 9	eqtri 2785	. 2 ⊢ (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11		df-pr 4591	. 2 ⊢ {(𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
12	6, 10, 11	3eqtr4g 2822	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹‘𝐵), (𝐹‘𝐶)})

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ∪ cun 3902  {csn 4588  {cpr 4590   “ cima 5663   Fn wfn 6531  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is used by:  fnimatpd  6965  fvinim0ffz  13825  mrcun  17684  neg1s  28231  dfpth2  30089  poimirlem1  38300  poimirlem9  38308  imarnf1pr  48047  uhgrimprop  48685  isuspgrimlem  48688  upgrimwlklem5  48694  upgrimpths  48702  clnbgrgrimlem  48726  clnbgrgrim  48727  grimgrtri  48742  isubgr3stgrlem4  48762  isubgr3stgrlem7  48765  grlimprclnbgr  48789  grlimgrtrilem2  48795