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Theorem funiun 7075
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 6514 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 dffn5 6884 . . 3 (𝐹 Fn dom 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
31, 2sylbb 218 . 2 (Fun 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
4 fvex 6838 . . 3 (𝐹𝑥) ∈ V
54dfmpt 7072 . 2 (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}
63, 5eqtrdi 2792 1 (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  {csn 4573  cop 4579   ciun 4941  cmpt 5175  dom cdm 5620  Fun wfun 6473   Fn wfn 6474  cfv 6479
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487
This theorem is referenced by:  funopsn  7076
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