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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 6406 | . . 3 ⊢ (Fun (𝐹 ∪ 𝐺) → Rel (𝐹 ∪ 𝐺)) | |
2 | relun 5690 | . . 3 ⊢ (Rel (𝐹 ∪ 𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺)) | |
3 | 1, 2 | sylib 221 | . 2 ⊢ (Fun (𝐹 ∪ 𝐺) → (Rel 𝐹 ∧ Rel 𝐺)) |
4 | simpl 486 | . . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐹) | |
5 | fununmo 6436 | . . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦) | |
6 | 5 | alrimiv 1935 | . . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐹𝑦) |
7 | 4, 6 | anim12i 616 | . . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) |
8 | dffun6 6404 | . . . 4 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | |
9 | 7, 8 | sylibr 237 | . . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐹) |
10 | simpr 488 | . . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐺) | |
11 | uncom 4076 | . . . . . . . 8 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹) | |
12 | 11 | funeqi 6410 | . . . . . . 7 ⊢ (Fun (𝐹 ∪ 𝐺) ↔ Fun (𝐺 ∪ 𝐹)) |
13 | fununmo 6436 | . . . . . . 7 ⊢ (Fun (𝐺 ∪ 𝐹) → ∃*𝑦 𝑥𝐺𝑦) | |
14 | 12, 13 | sylbi 220 | . . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐺𝑦) |
15 | 14 | alrimiv 1935 | . . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐺𝑦) |
16 | 10, 15 | anim12i 616 | . . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦)) |
17 | dffun6 6404 | . . . 4 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦)) | |
18 | 16, 17 | sylibr 237 | . . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐺) |
19 | 9, 18 | jca 515 | . 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 399 ∀wal 1541 ∃*wmo 2538 ∪ cun 3873 class class class wbr 5062 Rel wrel 5565 Fun wfun 6383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pr 5331 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3417 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-sn 4551 df-pr 4553 df-op 4557 df-br 5063 df-opab 5125 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-fun 6391 |
This theorem is referenced by: fsuppunbi 9019 |
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