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Theorem fununfun 6613
Description: If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.)
Assertion
Ref Expression
fununfun (Fun (𝐹𝐺) → (Fun 𝐹 ∧ Fun 𝐺))

Proof of Theorem fununfun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funrel 6582 . . 3 (Fun (𝐹𝐺) → Rel (𝐹𝐺))
2 relun 5820 . . 3 (Rel (𝐹𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺))
31, 2sylib 218 . 2 (Fun (𝐹𝐺) → (Rel 𝐹 ∧ Rel 𝐺))
4 simpl 482 . . . . 5 ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐹)
5 fununmo 6612 . . . . . 6 (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐹𝑦)
65alrimiv 1926 . . . . 5 (Fun (𝐹𝐺) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
74, 6anim12i 613 . . . 4 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
8 dffun6 6573 . . . 4 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
97, 8sylibr 234 . . 3 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → Fun 𝐹)
10 simpr 484 . . . . 5 ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐺)
11 uncom 4157 . . . . . . . 8 (𝐹𝐺) = (𝐺𝐹)
1211funeqi 6586 . . . . . . 7 (Fun (𝐹𝐺) ↔ Fun (𝐺𝐹))
13 fununmo 6612 . . . . . . 7 (Fun (𝐺𝐹) → ∃*𝑦 𝑥𝐺𝑦)
1412, 13sylbi 217 . . . . . 6 (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐺𝑦)
1514alrimiv 1926 . . . . 5 (Fun (𝐹𝐺) → ∀𝑥∃*𝑦 𝑥𝐺𝑦)
1610, 15anim12i 613 . . . 4 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
17 dffun6 6573 . . . 4 (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
1816, 17sylibr 234 . . 3 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → Fun 𝐺)
199, 18jca 511 . 2 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → (Fun 𝐹 ∧ Fun 𝐺))
203, 19mpancom 688 1 (Fun (𝐹𝐺) → (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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