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Theorem fununmo 6234
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 6204 . 2 (Fun (𝐹𝐺) → ∃*𝑦 𝑥(𝐹𝐺)𝑦)
2 orc 853 . . . 4 (𝑥𝐹𝑦 → (𝑥𝐹𝑦𝑥𝐺𝑦))
3 brun 4980 . . . 4 (𝑥(𝐹𝐺)𝑦 ↔ (𝑥𝐹𝑦𝑥𝐺𝑦))
42, 3sylibr 226 . . 3 (𝑥𝐹𝑦𝑥(𝐹𝐺)𝑦)
54moimi 2553 . 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 833  ∃*wmo 2545  cun 3828   class class class wbr 4929  Fun wfun 6182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-fun 6190
This theorem is referenced by:  fununfun  6235
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