Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnimatp Structured version   Visualization version   GIF version

Theorem fnimatp 31902
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 6976 . . . 4 ((𝐹 Fn 𝐷𝐴𝐷𝐵𝐷) → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
51, 2, 3, 4syl3anc 1372 . . 3 (𝜑 → (𝐹 “ {𝐴, 𝐵}) = {(𝐹𝐴), (𝐹𝐵)})
6 fnimatp.4 . . . . 5 (𝜑𝐶𝐷)
7 fnsnfv 6971 . . . . 5 ((𝐹 Fn 𝐷𝐶𝐷) → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
81, 6, 7syl2anc 585 . . . 4 (𝜑 → {(𝐹𝐶)} = (𝐹 “ {𝐶}))
98eqcomd 2739 . . 3 (𝜑 → (𝐹 “ {𝐶}) = {(𝐹𝐶)})
105, 9uneq12d 4165 . 2 (𝜑 → ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶})) = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)}))
11 df-tp 4634 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1211imaeq2i 6058 . . 3 (𝐹 “ {𝐴, 𝐵, 𝐶}) = (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶}))
13 imaundi 6150 . . 3 (𝐹 “ ({𝐴, 𝐵} ∪ {𝐶})) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
1412, 13eqtri 2761 . 2 (𝐹 “ {𝐴, 𝐵, 𝐶}) = ((𝐹 “ {𝐴, 𝐵}) ∪ (𝐹 “ {𝐶}))
15 df-tp 4634 . 2 {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)} = ({(𝐹𝐴), (𝐹𝐵)} ∪ {(𝐹𝐶)})
1610, 14, 153eqtr4g 2798 1 (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cun 3947  {csn 4629  {cpr 4631  {ctp 4633  cima 5680   Fn wfn 6539  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  s3rn  32112
  Copyright terms: Public domain W3C validator