Theorem funiun 5881

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 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})

Distinct variable group:   𝑥,𝐹

Proof of Theorem funiun

Step	Hyp	Ref	Expression
1		funfn 5402	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		dffn5im 5742	. . 3 ⊢ (𝐹 Fn dom 𝐹 → 𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)))
3	1, 2	sylbi 121	. 2 ⊢ (Fun 𝐹 → 𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)))
4		funfvex 5707	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
5	4	ralrimiva 2623	. . 3 ⊢ (Fun 𝐹 → ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) ∈ V)
6		dfmptg 5879	. . 3 ⊢ (∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) ∈ V → (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})
7	5, 6	syl 14	. 2 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})
8	3, 7	eqtrd 2271	1 ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821  {csn 3705  ⟨cop 3708  ∪ ciun 4007   ↦ cmpt 4187  dom cdm 4769  Fun wfun 5366   Fn wfn 5367  ‘cfv 5372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380

This theorem is referenced by: (None)