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Theorem fununmo 5847
 Description: If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.)
Assertion
Ref Expression
fununmo (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐹   𝑦,𝐺
Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem fununmo
StepHypRef Expression
1 funmo 5820 . 2 (Fun (𝐹𝐺) → ∃*𝑦 𝑥(𝐹𝐺)𝑦)
2 orc 399 . . . 4 (𝑥𝐹𝑦 → (𝑥𝐹𝑦𝑥𝐺𝑦))
3 brun 4633 . . . 4 (𝑥(𝐹𝐺)𝑦 ↔ (𝑥𝐹𝑦𝑥𝐺𝑦))
42, 3sylibr 223 . . 3 (𝑥𝐹𝑦𝑥(𝐹𝐺)𝑦)
54moimi 2508 . 2 (∃*𝑦 𝑥(𝐹𝐺)𝑦 → ∃*𝑦 𝑥𝐹𝑦)
61, 5syl 17 1 (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐹𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382  ∃*wmo 2459   ∪ cun 3538   class class class wbr 4583  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806 This theorem is referenced by:  fununfun  5848
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