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Theorem funiun 6558
Description: A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.)
Assertion
Ref Expression
funiun (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
Distinct variable group:   𝑥,𝐹

Proof of Theorem funiun
StepHypRef Expression
1 funfn 6060 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 dffn5 6385 . . 3 (𝐹 Fn dom 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
31, 2sylbb 209 . 2 (Fun 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
4 fvex 6344 . . 3 (𝐹𝑥) ∈ V
54dfmpt 6556 . 2 (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}
63, 5syl6eq 2821 1 (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  {csn 4317  cop 4323   ciun 4655  cmpt 4864  dom cdm 5250  Fun wfun 6024   Fn wfn 6025  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038
This theorem is referenced by:  funopsn  6559
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