Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > funiun | Structured version Visualization version GIF version |
Description: A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.) |
Ref | Expression |
---|---|
funiun | ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfn 6514 | . . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | dffn5 6884 | . . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ 𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥))) | |
3 | 1, 2 | sylbb 218 | . 2 ⊢ (Fun 𝐹 → 𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥))) |
4 | fvex 6838 | . . 3 ⊢ (𝐹‘𝑥) ∈ V | |
5 | 4 | dfmpt 7072 | . 2 ⊢ (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} |
6 | 3, 5 | eqtrdi 2792 | 1 ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) |
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 |
Copyright terms: Public domain | W3C validator |