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 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})

Distinct variable group:   𝑥,𝐹

Proof of Theorem funiun

Step	Hyp	Ref	Expression
1		funfn 6153	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		dffn5 6488	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ 𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)))
3	1, 2	sylbb 211	. 2 ⊢ (Fun 𝐹 → 𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)))
4		fvex 6446	. . 3 ⊢ (𝐹‘𝑥) ∈ V
5	4	dfmpt 6660	. 2 ⊢ (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}
6	3, 5	syl6eq 2877	1 ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})

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