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Theorem funiun 6664
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 6154 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 dffn5 6489 . . 3 (𝐹 Fn dom 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
31, 2sylbb 211 . 2 (Fun 𝐹𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
4 fvex 6447 . . 3 (𝐹𝑥) ∈ V
54dfmpt 6661 . 2 (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}
63, 5syl6eq 2878 1 (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  {csn 4398  cop 4404   ciun 4741  cmpt 4953  dom cdm 5343  Fun wfun 6118   Fn wfn 6119  cfv 6124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132
This theorem is referenced by:  funopsn  6665
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