Theorem fnimatpd 6993

Description: The image of an unordered triple under a function. (Contributed by Thierry Arnoux, 19-Sep-2023.)

Hypotheses

Ref	Expression
fnimatpd.1	⊢ (𝜑 → 𝐹 Fn 𝐷)
fnimatpd.2	⊢ (𝜑 → 𝐴 ∈ 𝐷)
fnimatpd.3	⊢ (𝜑 → 𝐵 ∈ 𝐷)
fnimatpd.4	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Assertion

Ref	Expression
fnimatpd	⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)})

Proof of Theorem fnimatpd

Step	Hyp	Ref	Expression
1		fnimatpd.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷)
2		fnimatpd.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
3		fnimatpd.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
4		fnimapr 6992	. . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
5	1, 2, 3, 4	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
6		fnimatpd.4	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
7		fnsnfv 6988	. . . . 5 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐶 ∈ 𝐷) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
8	1, 6, 7	syl2anc 584	. . . 4 ⊢ (𝜑 → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
9	8	eqcomd 2743	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)})
10	5, 9	uneq12d 4169	. 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
11		df-tp 4631	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
12	11	imaeq2i 6076	. . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13		imaundi 6169	. . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
14	12, 13	eqtri 2765	. 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15		df-tp 4631	. 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
16	10, 14, 15	3eqtr4g 2802	1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ∪ cun 3949  {csn 4626  {cpr 4628  {ctp 4630   “ 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-tp 4631  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:  s3rnOLD  32930  cycl3grtri  47914  grtrimap  47915