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Mirrors > Home > MPE Home > Th. List > fununmo | Structured version Visualization version GIF version |
Description: If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.) |
Ref | Expression |
---|---|
fununmo | ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmo 6204 | . 2 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥(𝐹 ∪ 𝐺)𝑦) | |
2 | orc 853 | . . . 4 ⊢ (𝑥𝐹𝑦 → (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦)) | |
3 | brun 4980 | . . . 4 ⊢ (𝑥(𝐹 ∪ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦)) | |
4 | 2, 3 | sylibr 226 | . . 3 ⊢ (𝑥𝐹𝑦 → 𝑥(𝐹 ∪ 𝐺)𝑦) |
5 | 4 | moimi 2553 | . 2 ⊢ (∃*𝑦 𝑥(𝐹 ∪ 𝐺)𝑦 → ∃*𝑦 𝑥𝐹𝑦) |
6 | 1, 5 | syl 17 | 1 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 833 ∃*wmo 2545 ∪ cun 3828 class class class wbr 4929 Fun wfun 6182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-br 4930 df-opab 4992 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-fun 6190 |
This theorem is referenced by: fununfun 6235 |
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