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Theorem fununmo 5931
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 5902 . 2 (Fun (𝐹𝐺) → ∃*𝑦 𝑥(𝐹𝐺)𝑦)
2 orc 400 . . . 4 (𝑥𝐹𝑦 → (𝑥𝐹𝑦𝑥𝐺𝑦))
3 brun 4701 . . . 4 (𝑥(𝐹𝐺)𝑦 ↔ (𝑥𝐹𝑦𝑥𝐺𝑦))
42, 3sylibr 224 . . 3 (𝑥𝐹𝑦𝑥(𝐹𝐺)𝑦)
54moimi 2519 . 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 383  ∃*wmo 2470  cun 3570   class class class wbr 4651  Fun wfun 5880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-fun 5888
This theorem is referenced by:  fununfun  5932
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