Theorem fnimapr 5760

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 5759	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
2	1	3adant3 1048	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
3		fnsnfv 5759	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
4	3	3adant2 1047	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
5	2, 4	uneq12d 3384	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})))
6	5	eqcomd 2244	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
7		df-pr 3715	. . . 4 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶})
8	7	imaeq2i 5122	. . 3 ⊢ (𝐹 “ {𝐵, 𝐶}) = (𝐹 “ ({𝐵} ∪ {𝐶}))
9		imaundi 5198	. . 3 ⊢ (𝐹 “ ({𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
10	8, 9	eqtri 2259	. 2 ⊢ (𝐹 “ {𝐵, 𝐶}) = ((𝐹 “ {𝐵}) ∪ (𝐹 “ {𝐶}))
11		df-pr 3715	. 2 ⊢ {(𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
12	6, 10, 11	3eqtr4g 2296	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹‘𝐵), (𝐹‘𝐶)})

Syntax hints:   → wi 4   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ∪ cun 3218  {csn 3708  {cpr 3709   “ cima 4775   Fn wfn 5370  ‘cfv 5375

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383

This theorem is referenced by:  en2  7106  fvinim0ffz  10643