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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 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)〉}) |
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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