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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 6378 | . . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | dffn5 6717 | . . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ 𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥))) | |
3 | 1, 2 | sylbb 220 | . 2 ⊢ (Fun 𝐹 → 𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥))) |
4 | fvex 6676 | . . 3 ⊢ (𝐹‘𝑥) ∈ V | |
5 | 4 | dfmpt 6898 | . 2 ⊢ (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) = ∪ 𝑥 ∈ dom 𝐹{〈𝑥, (𝐹‘𝑥)〉} |
6 | 3, 5 | syl6eq 2869 | 1 ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{〈𝑥, (𝐹‘𝑥)〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 {csn 4557 〈cop 4563 ∪ ciun 4910 ↦ cmpt 5137 dom cdm 5548 Fun wfun 6342 Fn wfn 6343 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: funopsn 6902 |
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