Theorem fnimatpd 6918

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 6917	. . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
5	1, 2, 3, 4	syl3anc 1379	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
6		fnimatpd.4	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
7		fnsnfv 6913	. . . . 5 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐶 ∈ 𝐷) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
8	1, 6, 7	syl2anc 590	. . . 4 ⊢ (𝜑 → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
9	8	eqcomd 2746	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)})
10	5, 9	uneq12d 4106	. 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
11		df-tp 4567	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
12	11	imaeq2i 6017	. . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13		imaundi 6107	. . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
14	12, 13	eqtri 2763	. 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15		df-tp 4567	. 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
16	10, 14, 15	3eqtr4g 2800	1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)})

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ∪ cun 3888  {csn 4562  {cpr 4564  {ctp 4566   “ cima 5628   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  s3rnOLD  33032  cycl3grtri  48439  grtrimap  48440