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Theorem fnimatp 30758
Description: The image of an unordered triple under a function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
Hypotheses
Ref Expression
fnimatp.1 (𝜑𝐹 Fn 𝐷)
fnimatp.2 (𝜑𝐴𝐷)
fnimatp.3 (𝜑𝐵𝐷)
fnimatp.4 (𝜑𝐶𝐷)
Assertion
Ref Expression
fnimatp (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})

Proof of Theorem fnimatp
StepHypRef Expression
1 fnimatp.1 . . . 4 (𝜑𝐹 Fn 𝐷)
2 fnimatp.2 . . . 4 (𝜑𝐴𝐷)
3 fnimatp.3 . . . 4 (𝜑𝐵𝐷)
4 fnimapr 6813 . . . 4 ((𝐹 Fn 𝐷𝐴𝐷𝐵𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
51, 2, 3, 4syl3anc 1373 . . 3 (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
6 fnimatp.4 . . . . 5 (𝜑𝐶𝐷)
7 fnsnfv 6808 . . . . 5 ((𝐹 Fn 𝐷𝐶𝐷) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
81, 6, 7syl2anc 587 . . . 4 (𝜑 → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
98eqcomd 2744 . . 3 (𝜑 → (𝐹 “ {𝐶}) = {(𝐹𝐶)})
105, 9uneq12d 4092 . 2 (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)}))
11 df-tp 4560 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1211imaeq2i 5941 . . 3 (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13 imaundi 6027 . . 3 (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
1412, 13eqtri 2766 . 2 (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15 df-tp 4560 . 2 {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)})
1610, 14, 153eqtr4g 2804 1 (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2111  cun 3878  {csn 4555  {cpr 4557  {ctp 4559  cima 5568   Fn wfn 6392  cfv 6397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-iota 6355  df-fun 6399  df-fn 6400  df-fv 6405
This theorem is referenced by:  s3rn  30964
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