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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 6365 | . . 3 ⊢ (Fun (𝐹 ∪ 𝐺) → Rel (𝐹 ∪ 𝐺)) | |
2 | relun 5677 | . . 3 ⊢ (Rel (𝐹 ∪ 𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺)) | |
3 | 1, 2 | sylib 219 | . 2 ⊢ (Fun (𝐹 ∪ 𝐺) → (Rel 𝐹 ∧ Rel 𝐺)) |
4 | simpl 483 | . . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐹) | |
5 | fununmo 6394 | . . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦) | |
6 | 5 | alrimiv 1919 | . . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐹𝑦) |
7 | 4, 6 | anim12i 612 | . . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) |
8 | dffun6 6363 | . . . 4 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | |
9 | 7, 8 | sylibr 235 | . . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐹) |
10 | simpr 485 | . . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐺) | |
11 | uncom 4126 | . . . . . . . 8 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹) | |
12 | 11 | funeqi 6369 | . . . . . . 7 ⊢ (Fun (𝐹 ∪ 𝐺) ↔ Fun (𝐺 ∪ 𝐹)) |
13 | fununmo 6394 | . . . . . . 7 ⊢ (Fun (𝐺 ∪ 𝐹) → ∃*𝑦 𝑥𝐺𝑦) | |
14 | 12, 13 | sylbi 218 | . . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐺𝑦) |
15 | 14 | alrimiv 1919 | . . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐺𝑦) |
16 | 10, 15 | anim12i 612 | . . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦)) |
17 | dffun6 6363 | . . . 4 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦)) | |
18 | 16, 17 | sylibr 235 | . . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐺) |
19 | 9, 18 | jca 512 | . 2 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Fun 𝐹 ∧ Fun 𝐺)) |
20 | 3, 19 | mpancom 684 | 1 ⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 ∃*wmo 2613 ∪ cun 3931 class class class wbr 5057 Rel wrel 5553 Fun wfun 6342 |
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-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 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-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-fun 6350 |
This theorem is referenced by: fsuppunbi 8842 |
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