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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 6154 | . . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | dffn5 6489 | . . 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 6447 | . . 3 ⊢ (𝐹‘𝑥) ∈ V | |
5 | 4 | dfmpt 6661 | . 2 ⊢ (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) = ∪ 𝑥 ∈ dom 𝐹{〈𝑥, (𝐹‘𝑥)〉} |
6 | 3, 5 | syl6eq 2878 | 1 ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{〈𝑥, (𝐹‘𝑥)〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 {csn 4398 〈cop 4404 ∪ ciun 4741 ↦ cmpt 4953 dom cdm 5343 Fun wfun 6118 Fn wfn 6119 ‘cfv 6124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 |
This theorem is referenced by: funopsn 6665 |
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