Theorem fnimatpd 6992

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 6991	. . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
5	1, 2, 3, 4	syl3anc 1399	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹‘𝐴), (𝐹‘𝐵)})
6		fnimatpd.4	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
7		fnsnfv 6987	. . . . 5 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐶 ∈ 𝐷) → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
8	1, 6, 7	syl2anc 596	. . . 4 ⊢ (𝜑 → {(𝐹‘𝐶)} = (𝐹 “ {𝐶}))
9	8	eqcomd 2769	. . 3 ⊢ (𝜑 → (𝐹 “ {𝐶}) = {(𝐹‘𝐶)})
10	5, 9	uneq12d 4171	. 2 ⊢ (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)}))
11		df-tp 4593	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
12	11	imaeq2i 6062	. . 3 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13		imaundi 6153	. . 3 ⊢ (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
14	12, 13	eqtri 2786	. 2 ⊢ (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15		df-tp 4593	. 2 ⊢ {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)} = ({(𝐹‘𝐴), (𝐹‘𝐵)} ∪ {(𝐹‘𝐶)})
16	10, 14, 15	3eqtr4g 2823	1 ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  {csn 4588  {cpr 4590  {ctp 4592   “ cima 5658   Fn wfn 6553  ‘cfv 6558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-fv 6566

This theorem is used by:  cycl3grtri  49250  grtrimap  49251