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Theorem fununmo 6588
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 6556 . 2 (Fun (𝐹𝐺) → ∃*𝑦 𝑥(𝐹𝐺)𝑦)
2 orc 864 . . . 4 (𝑥𝐹𝑦 → (𝑥𝐹𝑦𝑥𝐺𝑦))
3 brun 5192 . . . 4 (𝑥(𝐹𝐺)𝑦 ↔ (𝑥𝐹𝑦𝑥𝐺𝑦))
42, 3sylibr 233 . . 3 (𝑥𝐹𝑦𝑥(𝐹𝐺)𝑦)
54moimi 2533 . 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 844  ∃*wmo 2526  cun 3941   class class class wbr 5141  Fun wfun 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-fun 6538
This theorem is referenced by:  fununfun  6589
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