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| Mirrors > Home > MPE Home > Th. List > fununfun | Structured version Visualization version GIF version | ||
| Description: If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.) | 
| Ref | Expression | 
|---|---|
| fununfun | ⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | funrel 6582 | . . 3 ⊢ (Fun (𝐹 ∪ 𝐺) → Rel (𝐹 ∪ 𝐺)) | |
| 2 | relun 5820 | . . 3 ⊢ (Rel (𝐹 ∪ 𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺)) | |
| 3 | 1, 2 | sylib 218 | . 2 ⊢ (Fun (𝐹 ∪ 𝐺) → (Rel 𝐹 ∧ Rel 𝐺)) | 
| 4 | simpl 482 | . . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐹) | |
| 5 | fununmo 6612 | . . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦) | |
| 6 | 5 | alrimiv 1926 | . . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐹𝑦) | 
| 7 | 4, 6 | anim12i 613 | . . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | 
| 8 | dffun6 6573 | . . . 4 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | |
| 9 | 7, 8 | sylibr 234 | . . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐹) | 
| 10 | simpr 484 | . . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐺) | |
| 11 | uncom 4157 | . . . . . . . 8 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹) | |
| 12 | 11 | funeqi 6586 | . . . . . . 7 ⊢ (Fun (𝐹 ∪ 𝐺) ↔ Fun (𝐺 ∪ 𝐹)) | 
| 13 | fununmo 6612 | . . . . . . 7 ⊢ (Fun (𝐺 ∪ 𝐹) → ∃*𝑦 𝑥𝐺𝑦) | |
| 14 | 12, 13 | sylbi 217 | . . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐺𝑦) | 
| 15 | 14 | alrimiv 1926 | . . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐺𝑦) | 
| 16 | 10, 15 | anim12i 613 | . . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦)) | 
| 17 | dffun6 6573 | . . . 4 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦)) | |
| 18 | 16, 17 | sylibr 234 | . . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐺) | 
| 19 | 9, 18 | jca 511 | . 2 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Fun 𝐹 ∧ Fun 𝐺)) | 
| 20 | 3, 19 | mpancom 688 | 1 ⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1537 ∃*wmo 2537 ∪ cun 3948 class class class wbr 5142 Rel wrel 5689 Fun wfun 6554 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-fun 6562 | 
| This theorem is referenced by: fsuppunbi 9430 | 
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