Theorem fnimatpd 7006

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 7005	. . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
5	1, 2, 3, 4	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
6		fnimatpd.4	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
7		fnsnfv 7001	. . . . 5 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐶 ∈ 𝐷) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
8	1, 6, 7	syl2anc 583	. . . 4 ⊢ (𝜑 → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
9	8	eqcomd 2746	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)})
10	5, 9	uneq12d 4192	. 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
11		df-tp 4653	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
12	11	imaeq2i 6087	. . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13		imaundi 6181	. . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
14	12, 13	eqtri 2768	. 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15		df-tp 4653	. 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
16	10, 14, 15	3eqtr4g 2805	1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∪ cun 3974  {csn 4648  {cpr 4650  {ctp 4652   “ cima 5703   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  s3rnOLD  32912  grtrimap  47797