Theorem fnimapr 5702

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 5701	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
2	1	3adant3 1041	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
3		fnsnfv 5701	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
4	3	3adant2 1040	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
5	2, 4	uneq12d 3360	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
6	5	eqcomd 2235	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
7		df-pr 3674	. . . 4 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
8	7	imaeq2i 5072	. . 3 ⊢ (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9		imaundi 5147	. . 3 ⊢ (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
10	8, 9	eqtri 2250	. 2 ⊢ (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11		df-pr 3674	. 2 ⊢ {(𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
12	6, 10, 11	3eqtr4g 2287	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹‘𝐵), (𝐹‘𝐶)})

Syntax hints:   → wi 4   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   ∪ cun 3196  {csn 3667  {cpr 3668   “ cima 4726   Fn wfn 5319  ‘cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332

This theorem is referenced by:  en2  6993  fvinim0ffz  10480