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Theorem fnimatp 30407
Description: The image of a triplet 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 6721 . . . 4 ((𝐹 Fn 𝐷𝐴𝐷𝐵𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
51, 2, 3, 4syl3anc 1367 . . 3 (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
6 fnimatp.4 . . . . 5 (𝜑𝐶𝐷)
7 fnsnfv 6717 . . . . 5 ((𝐹 Fn 𝐷𝐶𝐷) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
81, 6, 7syl2anc 586 . . . 4 (𝜑 → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
98eqcomd 2826 . . 3 (𝜑 → (𝐹 “ {𝐶}) = {(𝐹𝐶)})
105, 9uneq12d 4116 . 2 (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)}))
11 df-tp 4546 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1211imaeq2i 5901 . . 3 (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13 imaundi 5982 . . 3 (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
1412, 13eqtri 2843 . 2 (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15 df-tp 4546 . 2 {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)})
1610, 14, 153eqtr4g 2880 1 (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  cun 3910  {csn 4541  {cpr 4543  {ctp 4545  cima 5532   Fn wfn 6324  cfv 6329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pr 5304
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3752  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-fv 6337
This theorem is referenced by:  s3rn  30606
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