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Theorem funiun 6663
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 6153 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 dffn5 6488 . . 3 (𝐹 Fn dom 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
31, 2sylbb 211 . 2 (Fun 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
4 fvex 6446 . . 3 (𝐹𝑥) ∈ V
54dfmpt 6660 . 2 (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}
63, 5syl6eq 2877 1 (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  {csn 4397  cop 4403   ciun 4740  cmpt 4952  dom cdm 5342  Fun wfun 6117   Fn wfn 6118  cfv 6123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131
This theorem is referenced by:  funopsn  6664
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