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Theorem fununfun 6550
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 6519 . . 3 (Fun (𝐹𝐺) → Rel (𝐹𝐺))
2 relun 5768 . . 3 (Rel (𝐹𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺))
31, 2sylib 217 . 2 (Fun (𝐹𝐺) → (Rel 𝐹 ∧ Rel 𝐺))
4 simpl 484 . . . . 5 ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐹)
5 fununmo 6549 . . . . . 6 (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐹𝑦)
65alrimiv 1931 . . . . 5 (Fun (𝐹𝐺) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
74, 6anim12i 614 . . . 4 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
8 dffun6 6510 . . . 4 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
97, 8sylibr 233 . . 3 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → Fun 𝐹)
10 simpr 486 . . . . 5 ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐺)
11 uncom 4114 . . . . . . . 8 (𝐹𝐺) = (𝐺𝐹)
1211funeqi 6523 . . . . . . 7 (Fun (𝐹𝐺) ↔ Fun (𝐺𝐹))
13 fununmo 6549 . . . . . . 7 (Fun (𝐺𝐹) → ∃*𝑦 𝑥𝐺𝑦)
1412, 13sylbi 216 . . . . . 6 (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐺𝑦)
1514alrimiv 1931 . . . . 5 (Fun (𝐹𝐺) → ∀𝑥∃*𝑦 𝑥𝐺𝑦)
1610, 15anim12i 614 . . . 4 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
17 dffun6 6510 . . . 4 (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
1816, 17sylibr 233 . . 3 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → Fun 𝐺)
199, 18jca 513 . 2 (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹𝐺)) → (Fun 𝐹 ∧ Fun 𝐺))
203, 19mpancom 687 1 (Fun (𝐹𝐺) → (Fun 𝐹 ∧ Fun 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  ∃*wmo 2533  cun 3909   class class class wbr 5106  Rel wrel 5639  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-fun 6499
This theorem is referenced by:  fsuppunbi  9331
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