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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 (𝜑𝐹 Fn 𝐷)
fnimatpd.2 (𝜑𝐴𝐷)
fnimatpd.3 (𝜑𝐵𝐷)
fnimatpd.4 (𝜑𝐶𝐷)
Assertion
Ref Expression
fnimatpd (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})

Proof of Theorem fnimatpd
StepHypRef Expression
1 fnimatpd.1 . . . 4 (𝜑𝐹 Fn 𝐷)
2 fnimatpd.2 . . . 4 (𝜑𝐴𝐷)
3 fnimatpd.3 . . . 4 (𝜑𝐵𝐷)
4 fnimapr 6992 . . . 4 ((𝐹 Fn 𝐷𝐴𝐷𝐵𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
51, 2, 3, 4syl3anc 1370 . . 3 (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
6 fnimatpd.4 . . . . 5 (𝜑𝐶𝐷)
7 fnsnfv 6988 . . . . 5 ((𝐹 Fn 𝐷𝐶𝐷) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
81, 6, 7syl2anc 584 . . . 4 (𝜑 → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
98eqcomd 2741 . . 3 (𝜑 → (𝐹 “ {𝐶}) = {(𝐹𝐶)})
105, 9uneq12d 4179 . 2 (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)}))
11 df-tp 4636 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1211imaeq2i 6078 . . 3 (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13 imaundi 6172 . . 3 (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
1412, 13eqtri 2763 . 2 (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15 df-tp 4636 . 2 {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)})
1610, 14, 153eqtr4g 2800 1 (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cun 3961  {csn 4631  {cpr 4633  {ctp 4635  cima 5692   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  s3rnOLD  32915  grtrimap  47851
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