Theorem fnimatp 30421

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

Hypotheses

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

Assertion

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

Proof of Theorem fnimatp

Step	Hyp	Ref	Expression
1		fnimatp.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷)
2		fnimatp.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
3		fnimatp.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
4		fnimapr 6740	. . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
5	1, 2, 3, 4	syl3anc 1366	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
6		fnimatp.4	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
7		fnsnfv 6736	. . . . 5 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐶 ∈ 𝐷) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
8	1, 6, 7	syl2anc 586	. . . 4 ⊢ (𝜑 → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
9	8	eqcomd 2826	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)})
10	5, 9	uneq12d 4133	. 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
11		df-tp 4565	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
12	11	imaeq2i 5920	. . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13		imaundi 6001	. . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
14	12, 13	eqtri 2843	. 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15		df-tp 4565	. 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
16	10, 14, 15	3eqtr4g 2880	1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)})

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113   ∪ cun 3927  {csn 4560  {cpr 4562  {ctp 4564   “ cima 5551   Fn wfn 6343  ‘cfv 6348

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356

This theorem is referenced by:  s3rn  30620