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Theorem fnimatp 32169
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 6974 . . . 4 ((𝐹 Fn 𝐷𝐴𝐷𝐵𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
51, 2, 3, 4syl3anc 1369 . . 3 (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
6 fnimatp.4 . . . . 5 (𝜑𝐶𝐷)
7 fnsnfv 6969 . . . . 5 ((𝐹 Fn 𝐷𝐶𝐷) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
81, 6, 7syl2anc 582 . . . 4 (𝜑 → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
98eqcomd 2736 . . 3 (𝜑 → (𝐹 “ {𝐶}) = {(𝐹𝐶)})
105, 9uneq12d 4163 . 2 (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)}))
11 df-tp 4632 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1211imaeq2i 6056 . . 3 (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13 imaundi 6148 . . 3 (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
1412, 13eqtri 2758 . 2 (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15 df-tp 4632 . 2 {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)})
1610, 14, 153eqtr4g 2795 1 (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2104  cun 3945  {csn 4627  {cpr 4629  {ctp 4631  cima 5678   Fn wfn 6537  cfv 6542
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  s3rn  32379
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