![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6517 | . 2 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥(𝐹 ∪ 𝐺)𝑦) | |
2 | orc 866 | . . . 4 ⊢ (𝑥𝐹𝑦 → (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦)) | |
3 | brun 5157 | . . . 4 ⊢ (𝑥(𝐹 ∪ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦)) | |
4 | 2, 3 | sylibr 233 | . . 3 ⊢ (𝑥𝐹𝑦 → 𝑥(𝐹 ∪ 𝐺)𝑦) |
5 | 4 | moimi 2540 | . 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 846 ∃*wmo 2533 ∪ cun 3909 class class class wbr 5106 Fun wfun 6491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-fun 6499 |
This theorem is referenced by: fununfun 6550 |
Copyright terms: Public domain | W3C validator |