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Theorem funiun 6366
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 5877 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 dffn5 6198 . . 3 (𝐹 Fn dom 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
31, 2sylbb 209 . 2 (Fun 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
4 fvex 6158 . . 3 (𝐹𝑥) ∈ V
54dfmpt 6364 . 2 (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}
63, 5syl6eq 2671 1 (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  {csn 4148  cop 4154   ciun 4485  cmpt 4673  dom cdm 5074  Fun wfun 5841   Fn wfn 5842  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  funopsn  6367
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