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Theorem fununfun 6437
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 6406 . . 3 (Fun (𝐹𝐺) → Rel (𝐹𝐺))
2 relun 5690 . . 3 (Rel (𝐹𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺))
31, 2sylib 221 . 2 (Fun (𝐹𝐺) → (Rel 𝐹 ∧ Rel 𝐺))
4 simpl 486 . . . . 5 ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐹)
5 fununmo 6436 . . . . . 6 (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐹𝑦)
65alrimiv 1935 . . . . 5 (Fun (𝐹𝐺) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
74, 6anim12i 616 . . . 4 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
8 dffun6 6404 . . . 4 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
97, 8sylibr 237 . . 3 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → Fun 𝐹)
10 simpr 488 . . . . 5 ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐺)
11 uncom 4076 . . . . . . . 8 (𝐹𝐺) = (𝐺𝐹)
1211funeqi 6410 . . . . . . 7 (Fun (𝐹𝐺) ↔ Fun (𝐺𝐹))
13 fununmo 6436 . . . . . . 7 (Fun (𝐺𝐹) → ∃*𝑦 𝑥𝐺𝑦)
1412, 13sylbi 220 . . . . . 6 (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐺𝑦)
1514alrimiv 1935 . . . . 5 (Fun (𝐹𝐺) → ∀𝑥∃*𝑦 𝑥𝐺𝑦)
1610, 15anim12i 616 . . . 4 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
17 dffun6 6404 . . . 4 (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
1816, 17sylibr 237 . . 3 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → Fun 𝐺)
199, 18jca 515 . 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 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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